{"id":2916,"date":"2023-12-19T11:29:02","date_gmt":"2023-12-19T05:59:02","guid":{"rendered":"https:\/\/envisionmathanswerkey.com\/?p=2916"},"modified":"2023-12-20T09:27:54","modified_gmt":"2023-12-20T03:57:54","slug":"envision-math-common-core-grade-7-answer-key-topic-8","status":"publish","type":"post","link":"https:\/\/envisionmathanswerkey.com\/envision-math-common-core-grade-7-answer-key-topic-8\/","title":{"rendered":"enVision Math Common Core Grade 7 Answer Key Topic 8 Solve Problems Involving Geometry"},"content":{"rendered":"

Go through the\u00a0enVision Math Common Core Grade 7 Answer Key<\/a>\u00a0Topic 8 Solve Problems Involving Geometry<\/strong>\u00a0regularly and improve your accuracy in solving questions.<\/p>\n

enVision Math Common Core 7th Grade Answers Key Topic 8 Solve Problems Involving Geometry<\/h2>\n

GET READY!
\nReview What You Know!<\/strong><\/p>\n

Vocabulary
\nChoose the best term from the box to complete each definition.
\n\"Envision<\/p>\n

Question 1.
\nThe number of square units that a figure covers is its _________.
\nAnswer:
\nWe know that,
\nThe number of square units that a figure covers is its “Area”
\nHence, from the above,
\nWe can conclude that the best term to complete the given definition is the “Area”<\/p>\n

Question 2.
\nThe _________ of a triangle is the length of the perpendicular line segment from a vertex to the opposite side.
\nAnswer:
\nWe know that,
\nThe “Height” of a triangle is the length of the perpendicular line segment from a vertex to the opposite side.
\nHence, from the above,
\nWe can conclude that the best term to complete the given definition is the “Height”<\/p>\n

Question 3.
\nThe _________ of a solid figure is the number of cubic units needed to fill it.
\nAnswer:
\nWe know that,
\nThe “Volume” of a solid figure is the number of cubic units needed to fill it
\nHence, from the above,
\nWe can conclude that the best term to complete the given definition is the “Volume”<\/p>\n

Question 4.
\nAny line segment that connects the center of a circle to a point on the circle is called a __________.
\nAnswer:
\nWe know that,
\nAny line segment that connects the center of a circle to a point on the circle is called a “Radius”
\nHence, from the above,
\nWe can conclude that the best term to complete the given definition is the “Radius”<\/p>\n

Area and Volume<\/strong>
\nFind each measure.<\/p>\n

Question 5.
\nArea of a triangle with a base 6 feet and height 9 feet
\nAnswer:
\nIt is given that
\nThe base of the triangle is: 6 feet
\nThe height of the triangle is: 9 feet
\nNow,
\nWe know that,
\nThe area of a triangle (A) = \\(\\frac{1}{2}\\) \u00d7 Base of the triangle \u00d7 Height of the triangle
\nSo,
\nA = \\(\\frac{1}{2}\\) \u00d7 6 \u00d7 9
\n= \\(\\frac{54}{2}\\)
\n= 27 feet\u00b2
\nHence, from the above,
\nWe can conclude that the area of the given triangle is: 27 feet\u00b2<\/p>\n

Question 6.
\nThe volume of a rectangular prism with length 4 inches, width 2 inches, and height 2 inches
\nAnswer:
\nIt is given that
\nThe length of a rectangular prism is: 4 inches
\nThe width of a rectangular prism is: 2 inches
\nThe height of a rectangular prism is: 2 inches
\nNow,
\nWe know that,
\nThe volume of a rectangular prism (V) = Length \u00d7 Width \u00d7 Height
\nSo,
\nV = 4 \u00d7 2 \u00d7 2
\n= 16 inches\u00b3
\nHence, from the above,
\nWe can conclude that the volume of the given rectangular prism is: 16 inches\u00b3<\/p>\n

Measure Angles<\/strong>
\nUse a protractor to find the measure of each angle.
\nQuestion 7.
\n\"Envision
\nAnswer:
\nTo measure the angle, use “https:\/\/www.geogebra.org\/calculator”
\nSo,
\nThe measured angle is:
\n\"\"
\nHence, from the above,
\nWe can conclude that the measured angle using the protractor is: 54.5\u00b0<\/p>\n

Question 8.
\n\"Envision
\nAnswer:
\nTo measure the angle, use “https:\/\/www.geogebra.org\/calculator”
\nSo,
\nThe measured angle is:
\n\"\"
\nHence, from the above,
\nWe can conclude that the measured angle using the protractor is: 104.3\u00b0<\/p>\n

Describe Characteristics of Shapes
\nDescribe this figure using as many geometry terms as you can.
\nQuestion 9.
\n\"Envision
\nAnswer:
\nThe given figure is:
\n\"Envision
\nNow,
\nFrom the given figure,
\nWe can observe that
\na. All the angles are 90\u00b0
\nb. The parallel sides are the same
\nc. The diagonals bisect each other
\nd. The lengths of the diagonals are equal
\nSo,
\n\"\"
\nHence, from the above,
\nWe can conclude that the given figure is a “Rectangle”<\/p>\n

\"Involving<\/p>\n

Language Development<\/strong>
\nFill in the word web to connect keywords you learn in this topic. A sample keyword and its connections are given.
\n\"Envision
\n\"Envision
\nPICK A PROJECT<\/strong><\/p>\n

PROJECT 8A
\nIf you built a sculpture, what materials would you use?
\nPROJECT: CONSTRUCT A THREE-DIMENSIONAL SCULPTURE<\/p>\n

PROJECT 8B
\nIf you made a pizza, what kind of pizza would it be?
\nPROJECT: ANALYZE A PEPPERONI PIZZA<\/p>\n

PROJECT 8C
\nWhat places have you visited where being a tour guide would be fun?
\nPROJECT: PLAN A GUIDED TOUR<\/p>\n

PROJECT 8D
\nHow could you determine which is a larger-a tall building or a wide building?
\nPROJECT: BUILD A SCALE MODEL<\/p>\n

Lesson 8.1 Solve Problems Involving Scale Drawings<\/h3>\n

Explore It!<\/strong><\/p>\n

Calvin made a scale model of the plane shown.
\n\"Envision<\/p>\n

I can… use the key in a scale drawing to find missing measures.<\/p>\n

A. How can you represent the relationship between the model of the plane and the actual plane?
\nAnswer:
\nThe given figure is:
\n\"Envision
\nNow,
\nThe representation of the relationship between the model of the plane and the actual plane is:
\nThe actual diagram is an enlarged diagram of the scale diagram
\nSo,
\nThe scale factor = \\(\\frac{15 in.}{240 ft}\\)
\n= \\(\\frac{1}{16}\\)
\nHence, from the above,
\nWe can conclude that the actual diagram is 16 times larger than the scale diagram<\/p>\n

B. What do you notice about the relationship between the model of the plane and the actual plane?
\nAnswer:
\nWe know that,
\nThe relationship between the model of the plane and the actual plane can be given by a “Scale factor”
\nSo,
\nThe scale factor = \\(\\frac{15 in.}{240 ft}\\)
\n= \\(\\frac{1}{16}\\)
\nHence, from the above,
\nWe can conclude that the actual diagram is 16 times larger than the scale diagram<\/p>\n

Focus on math practices<\/strong>
\nLook for Relationships<\/strong> If the model and the actual plane are to scale, what do you know about the relationship between all the other parts of the model and the actual plane, aside from the total length?
\nAnswer:
\nIf the model and the actual plane are to scale, then
\nAll the other parts of the model and the actual plane, aside from the total length are also scaled<\/p>\n

Essential Question<\/strong>
\nHow do scale drawings and actual measurements represent proportional relationships?
\nAnswer:
\nThe “Scale factor” is the amount that an image is enlarged or shrunk. It is also called ‘k’ because k represents the multiplier between two proportional ratios and scale drawings represent proportional relationships<\/p>\n

Try It!<\/strong><\/p>\n

What is the actual width, w, of the island if the width in the drawing is 2.5 inches?
\n\"Envision
\nThe actual width of the island is _______ feet.
\nAnswer:
\nFrom Example 1,
\nIt is given that,
\nScale: 1 inch = 1.5 ft
\nNow,
\n\\(\\frac{1.5}{1}\\) = \\(\\frac{w}{2.5}\\)
\nw = \\(\\frac{1.5 \u00d7 2.5}{1}\\)
\nw = 3.75 feet
\nHence, from the above,
\nWe can conclude that the actual width of the island is: 3.75 feet<\/p>\n

Convince Me!<\/strong> How would the proportion for Example 1 change if the scale changed?
\nAnswer:
\nFor Example 1,
\nIf the scale changed, then
\nThe proportion would increase or decrease
\nCase 1:
\nIf the scale changed to:
\n1 in = 2 ft
\nThen, the proportion will change like
\n\\(\\frac{2}{1}\\) = \\(\\frac{x}{6}\\)
\nx = \\(\\frac{6 \u00d7 2}{1}\\)
\nx = 12 feet
\nSo,
\nThe actual width of the island is: 12 feet
\nCase 2:
\nIf the scale changed to:
\n1 in = 1.25 ft
\nThen, the proportion will change like
\n\\(\\frac{1.25}{1}\\) = \\(\\frac{x}{6}\\)
\nx = \\(\\frac{6 \u00d7 1.25}{1}\\)
\nx = 7.5 feet
\nSo,
\nThe actual width of the island is: 7.5 feet
\nHence, from the above,
\nWe can conclude that
\nIf the scale increases, then the proportion also increases
\nIf the scale decreases, then the proportion also decreases<\/p>\n

Try It!<\/strong><\/p>\n

The scale drawing shown represents an existing barn. The shortest side of the barn measures 150 meters. If a new barn that is \\(\\frac{2}{3}\\) its size replaces the existing barn, what will be the scale of this drawing to the new barn?
\n\"Envision
\nAnswer:
\nIt is given that
\nThe scale drawing shown represents an existing barn. The shortest side of the barn measures 150 meters and a new barn that is \\(\\frac{2}{3}\\) its size replaces the existing barn
\nNow,
\nThe representation of the existing barn is:
\n\"Envision
\nNow,
\nThe length of the shortest side for a new barn = \\(\\frac{2}{3}\\) \u00d7 150
\n= 100 in.
\nSo,
\nThe scale factor for the shortest side for a new barn = \\(\\frac{150}{100}\\)
\n= 1.5
\nHence, from the above,
\nWe can conclude that
\nThe scale of the drawing to the new barn is 1: 1.5<\/p>\n

KEY CONCEPT<\/strong>
\nThe scale factor of a scale drawing is the ratio of an actual length, y, to the corresponding length, x, in the drawing. The ratio is the constant of proportionality, k, that relates the actual figure to the scale drawing. You can use a proportion or use an equation of the form y = kx to solve problems involving scale drawings.
\n\"Envision<\/p>\n

Do You Understand?<\/strong><\/p>\n

Question 1.
\nEssential Question<\/strong> How do scale drawings and actual measurements represent proportional relationships?
\nAnswer:
\nThe “Scale factor” is the amount that an image is enlarged or shrunk. It is also called “k” because k represents the multiplier between two proportional ratios and scale drawings represent proportional relationships<\/p>\n

Question 2.
\nGeneralize<\/strong> Describe the ratio of corresponding measures in scale drawings and the actual measures they represent?
\nAnswer:
\nA scale drawing is a drawing that is to scale an original image. When an object is “to scale,” that means it is the same shape, but not the same size. It is also called k because k represents the multiplier between two proportional ratios\u00a0and scale drawings represent proportional relationships.<\/p>\n

Question 3.
\nReasoning<\/strong> Mikayla is determining the actual distance between Harrisville and Lake Town using a map. The scale on her map reads. 1 inch = 50 miles. She measures the distance to be 4.5 inches and writes the following proportion:
\n\\(\\frac{1 \\mathrm{in.}}{4.5 \\mathrm{in.}}=\\frac{50 \\mathrm{mi}}{x \\mathrm{mi}}\\)
\nExplain why her proportion is equivalent to
\n\\(\\frac{50 \\mathrm{mi}}{1 \\text { in. }}=\\frac{x \\mathrm{mi}}{4.5 \\mathrm{in} .}\\)
\nAnswer:
\nIt is given that
\nReasoning Mikayla is determining the actual distance between Harrisville and Lake Town using a map. The scale on her map reads. 1 inch = 50 miles. She measures the distance to be 4.5 inches
\nSo,
\nThe scale for the distance between Harrisville and Lake Town using a map is: 50: 1
\nNow,
\nLet the value that is equivalent to 4.5 inches be x miles
\nNow,
\nTo find the value in miles that is equivalent to 4.5 inches,
\nWe have to write
\n\\(\\frac{1 in.}{4.5 in.}\\) = \\(\\frac{50 mi}{x mi}\\)
\nNow,
\nCross-Multiply the given equation
\nSo,
\n\\(\\frac{50 mi}{1 in.}\\) = \\(\\frac{x miles}{4.5 in.}\\)
\nHence, from the above,
\nWe can conclude that the proportion is in the above form because of the cross-multiplication<\/p>\n

Do You Know How?<\/strong><\/p>\n

Question 4.
\nWhat is the actual base length of the triangle depicted in the scale drawing?
\n\"Envision
\nAnswer:
\nThe given figure is:
\n\"Envision
\nNow,
\nFrom the given figure,
\nWe can observe that
\nThe scale factor = \\(\\frac{2}{1}\\)
\nSo,
\nThe actual base length of the given triangle = (The base length as depicted in the given triangle) \u00d7 (Scale factor)
\n= 2.5 \u00d7 \\(\\frac{2}{1}\\)
\n= 2.5 \u00d7 2
\n= 5 m.
\nHence, from the above,
\nWe can conclude that
\nThe actual base length of the triangle as depicted in the given scale drawing is: 5 m<\/p>\n

Question 5.
\nWhat is the area of the actual square window shown in the scale drawing?
\n\"Envision
\nAnswer:
\nThe given figure is:
\n\"Envision
\nNow,
\nFrom the given figure,
\nWe can observe that
\nScale factor = \\(\\frac{2}{1}\\)
\nSo,
\nThe actual side length of the given square window = (The side length of the given square window) \u00d7 (Scale factor)
\n= 0.75 \u00d7 2
\n= 15 m
\nNow,
\nWe know that,
\nThe area of a square = Side\u00b2
\nSo,
\nThe area of the actual square window = (Actual Side length of the given square window)\u00b2
\n= 15 m \u00d7 15 m
\n= 225 m\u00b2
\nHence, from the above,
\nWe can conclude that the area of the actual square window is: 225 m\u00b2<\/p>\n

Question 6.
\nA distance of 30 miles on a map is represented by a 2-inch line. If the map is enlarged to 3 times its size, what will be the scale of the enlarged map?
\nAnswer:
\nIt is given that
\nA distance of 30 miles on a map is represented by a 2-inch line and the map is enlarged to 3 times its size
\nNow,
\nThe scale distance on a map = \\(\\frac{Actual distance on a map}{Scale of a map}\\)
\n= \\(\\frac{30}{2}\\)
\n= 15 inches
\nNow,
\nThe scale distance on the enlarged map = (The scale distance on a normal map) \u00d7 3
\n= 15 \u00d7 3
\n= 45 inches
\nSo,
\nThe scale of the enlarged map = \\(\\frac{45}{3}\\)
\n= 15
\nHence, from the above,
\nWe can conclude that the scale of the enlarged map is: 1inch = 15 miles<\/p>\n

Practice & Problem Solving<\/strong><\/p>\n

Leveled Practice For 7 and 8, fill in the boxes to find the actual measures.
\nQuestion 7.
\nOn a map, 1 inch equals 5 miles. Two cities are 8 inches apart on the map.
\nWhat is the actual distance between the cities?
\n\"Envision
\nAnswer:
\nIt is given that
\nOn a map, 1 inch equals 5 miles. Two cities are 8 inches apart on the map.
\nNow,
\nThe scale factor from the given information is:
\n1 inch = 5 miles
\nSo,
\n\"\"
\nHence, from the above,
\nWe can conclude that
\nThe actual distance between the cities is: 40 miles<\/p>\n

Question 8.
\nRyan makes a scale drawing of a banner for a school dance. He uses a scale of 1 inch = 3 feet, and the width of the drawing is 5 inches. What is the actual width, w, of the banner?
\n\"Envision
\nAnswer:
\nIt is given that
\nRyan makes a scale drawing of a banner for a school dance. He uses a scale of 1 inch = 3 feet, and the width of the drawing is 5 inches
\nNow,
\nFrom the given information,
\nWe can observe that
\nThe actual width of the banner (w) = (The width of the drawing) \u00d7 (Scale factor)
\nSo,
\n\"\"
\nHence, from the above,
\nWe can conclude that
\nThe actual width (w) of the banner is: 15 feet<\/p>\n

Question 9.
\nOn a map, 1 inch equals 7.2 miles. Two houses are 1.5 inches apart on the map. What is the actual distance between the houses?
\nAnswer:
\nIt is given that
\nOn a map, 1 inch equals 7.2 miles. Two houses are 1.5 inches apart on the map
\nSo,
\nFrom the given information,
\nThe scale factor = \\(\\frac{7.2}{1}\\)
\nNow,
\nLet the actual distance between the houses be d
\nSo,
\n\\(\\frac{7.2}{1}\\) = \\(\\frac{d}{1.5}\\)
\nd = 7.2 \u00d7 1.5
\nd = 10.80 miles
\nHence, from the above,
\nWe can conclude that
\nThe actual distance between the houses is: 10.8 miles<\/p>\n

Question 10.
\nThe original blueprint for the Morenos’ living room has a scale of 2 inches = 5 feet. The family wants to use a new blueprint that shows the length of the living room to be 15 inches. If the width of the living room on the original blueprint is 6 inches and the length is 9.6 inches, what are the scale and the width of the new blueprint?
\nAnswer:
\nIt is given that
\nThe original blueprint for the Morenos’ living room has a scale of 2 inches = 5 feet. The family wants to use a new blueprint that shows the length of the living room to be 15 inches and the width of the living room on the original blueprint is 6 inches and the length is 9.6 inches
\nNow,
\nThe length of the new blueprint = (Scale factor) \u00d7 (The length of the original blueprint)
\n= \\(\\frac{5}{2}\\) \u00d7 9.6
\n= 5 \u00d7 4.8
\n= 24.0 inches
\n= 24 inches
\nThe width of the new blueprint = (Scale factor) \u00d7 (The width of the original blueprint)
\n= \\(\\frac{5}{2}\\) \u00d7 6
\n= 5 \u00d7 3
\n= 15 inches
\nNow,
\n\\(\\frac{24}{15}\\) = \\(\\frac{x}{1}\\)
\nx = \\(\\frac{8}{3}\\)
\nHence, from the above,
\nWe can conclude that
\nThe scale factor for the new blueprint is:
\n3 inches = 8 feet<\/p>\n

Question 11.
\nThe scale for a drawing of the tennis court is 1 centimeter = 2 meters. What is the area of the actual tennis court?
\n\"Envision
\nAnswer:
\nIt is given that
\nThe scale for a drawing of the tennis court is 1 centimeter = 2 meters.
\nNow,
\nThe given figure is:
\n\"Envision
\nSo,
\nThe length of the actual tennis court = (Scale factor) \u00d7 (The length of the scaled tennis court)
\n= 2 \u00d7 10
\n= 20 meters
\nSo,
\nThe width of the actual tennis court = (Scale factor) \u00d7 (The width of the scaled tennis court)
\n= 2 \u00d7 5.5
\n= 11 meters
\nNow,
\nWe know that,
\nThe area of a rectangle = Length \u00d7 Width
\nSo,
\nThe area of the actual tennis court = (The length of the actual tennis court) \u00d7 (The width of the actual tennis court)
\n= 20 \u00d7 11
\n= 220 meters\u00b2
\nHence, from the above,
\nWe can conclude that the area of the actual tennis court is: 220 meters\u00b2<\/p>\n

Question 12.
\nThe scale for the drawing of a rectangular playing field is 2 inches = 5 feet.
\na. Write an equation you can use to find the dimensions of the actual field, where x is a dimension of the scale drawing (in inches) and y is the corresponding dimension of the actual field (in feet).
\n\"Envision
\nAnswer:
\nIt is given that
\nThe scale for the drawing of a rectangular playing field is 2 inches = 5 feet.
\nNow,
\nThe given figure is:
\n\"Envision
\nNow,
\nFrom the given information,
\nThe scale factor = \\(\\frac{5}{2}\\)
\nSo,
\nThe actual length of the rectangular playing field = (Scale factor) \u00d7 (The length of the scaled rectangular playing field)
\n= \\(\\frac{5}{2}\\) \u00d7 20
\n= 5 \u00d7 10
\n= 50 feet
\nThe actual width of the rectangular playing field = (Scale factor) \u00d7 (The width of the scaled rectangular playing field)
\n= \\(\\frac{5}{2}\\) \u00d7 10
\n= 5 \u00d7 5
\n= 25 feet
\nSo,
\nThe scale factor for the actual rectangular playing field can be found out by:
\n\\(\\frac{50}{20}\\) = \\(\\frac{x}{1}\\)\u00a0 \u00a0(or)\u00a0 \u00a0\\(\\frac{25}{10}\\) = \\(\\frac{x}{1}\\)
\nx = \\(\\frac{5}{2}\\)
\nHence, from the above,
\nWe can conclude that
\nThe scale for the actual rectangular playing field is:
\n2 inches = 5 feet<\/p>\n

b. What is the area of the field?
\nAnswer:
\nFrom part (a),
\nWe can observe that
\nThe actual length of the rectangular playing field = (Scale factor) \u00d7 (The length of the scaled rectangular playing field)
\n= \\(\\frac{5}{2}\\) \u00d7 20
\n= 5 \u00d7 10
\n= 50 feet
\nThe actual width of the rectangular playing field = (Scale factor) \u00d7 (The width of the scaled rectangular playing field)
\n= \\(\\frac{5}{2}\\) \u00d7 10
\n= 5 \u00d7 5
\n= 25 feet
\nSo,
\nThe area of the rectangular playing field = (Actual length) \u00d7 (Actual width)
\n= 50 \u00d7 25
\n= 1,250 feet\u00b2
\nHence, from the above,
\nWe can conclude that
\nThe area of the field is: 1,250 feet\u00b2<\/p>\n

\"Involving<\/p>\n

Question 13.
\nHow many square feet of flooring are needed to cover the entire floor of Bedroom 1?
\n\"Envision
\nAnswer:<\/p>\n

Question 14.
\nThe actual distance between Point A and Point B is 200 meters. A length of 1.9 feet represents this distance on a certain wall map. Point C and Point D are 3.8 feet apart on this map. What is the actual distance between Point C and Point D?
\n\"Envision
\nAnswer:
\nIt is given that
\nThe actual distance between Point A and Point B is 200 meters. A length of 1.9 feet represents this distance on a certain wall map. Point C and Point D are 3.8 feet apart on this map
\nNow,
\nLet the actual distance between points C and D be x meters
\nSo,
\n\\(\\frac{200}{x}\\) = \\(\\frac{1.9}{3.8}\\)
\n\\(\\frac{200}{x}\\) = \\(\\frac{1}{2}\\)
\nx = 200 \u00d7 2
\nx = 400 meters
\nHence, from the above,
\nWe can conclude that
\nThe actual distance between points C and D is: 400 meters<\/p>\n

Question 15.
\nHigher-Order Thinking<\/strong> A map of a highway has a scale of 2 inches equals 33 miles. The length of the highway on the map is 6 inches. There are 11 rest stops equally spaced on the highway, including one at each end. You are making a new map with a scale of 1 inch equals 30 miles. How far apart are the rest stops on the new map?
\nAnswer:
\nIt is given that
\nA map of a highway has a scale of 2 inches equals 33 miles. The length of the highway on the map is 6 inches. There are 11 rest stops equally spaced on the highway, including one at each end. You are making a new map with a scale of 1 inch equals 30 miles
\nSo,
\nThe actual length of the highway according to the scale of 2 in = 33 miles is:
\nThe actual length of the highway = (Scale factor) \u00d7 (The length of the highway on the map)
\n= \\(\\frac{33}{2}\\) \u00d7 6
\n= 33 \u00d73
\n= 99 miles
\nThe actual length of the highway according to the scale of 1 in = 30 miles is:
\nThe actual length of the highway = (Scale factor) \u00d7 (The length of the highway on the map)
\n= \\(\\frac{30}{1}\\) \u00d7 6
\n= 30 \u00d76
\n= 180 miles
\nNow,
\nLet the number of stops on the highway with a scale of 1 in = 30 miles be x
\nSo,
\n\\(\\frac{99}{11}\\) = \\(\\frac{180}{x}\\)
\n\\(\\frac{180}{x}\\) = 9
\nx = \\(\\frac{180}{9}\\)
\nx = 20 stops
\nNow,
\nThe distance between the rest stops = \\(\\frac{30}{20}\\)
\n= 1.5 feet
\nHence, from the above,
\nWe can conclude that
\nThe distance between the rest stops on the new map are 1.5 feet<\/p>\n

Assessment Practice<\/strong>
\nQuestion 16.
\nThe original blueprint of a concrete patio has a scale of 2 inches = 3 feet.
\n\"Envision
\nVictoria wants to make a new blueprint of the patio with a length of 16.8 inches.
\nPART A
\nWhat is the scale for the new blueprint?
\n1 inch = _______ feet
\nAnswer:
\nIt is given that
\nThe original blueprint of a concrete patio has a scale of 2 inches = 3 feet.
\nNow,
\nThe given figure is:
\n\"Envision
\nNow,
\nFrom the given information,
\nScale factor = \\(\\frac{3}{2}\\)
\nSo,
\nThe length of the new blueprint = (Scale factor) \u00d7 (The length of the original blueprint)
\n= \\(\\frac{3}{2}\\) \u00d7 14
\n= 3 \u00d7 7
\n= 21 inches
\nNow,
\n\\(\\frac{14}{16.8}\\) = \\(\\frac{21}{x}\\)
\nx = 8.4 \u00d7 3
\nx = 25.2
\nHence, from the above,
\nWe can conclude that
\nThe scale factor for the new blueprint is:
\n1 inch = 25.2 feet<\/p>\n

PART B
\nWhat is the width, in inches, of the blueprint with the new scale?
\nAnswer:
\nThe given figure is:
\n\"Envision
\nSo,
\nThe width of the new blueprint = (Scale factor) \u00d7 (The length of the original blueprint)
\n= \\(\\frac{3}{2}\\) \u00d7 12
\n= 3 \u00d7 6
\n= 18 inches
\nHence, from the above,
\nWe can conclude that
\nThe width (w) of the blueprint with the new scale is: 18 inches<\/p>\n

Lesson 8.2 Draw Geometric Figures<\/h3>\n

Solve & Discuss It!<\/strong><\/p>\n

Students in the Art Club are designing a flag with the school’s mascot and emblem. The flag has four sides, with two sides that are twice as long as the other two sides. What shape could the flag be, and what dimensions could it have? Make and label a scale drawing as part of your answer.
\n\"Envision
\nI can… draw figures with given conditions.
\nAnswer:
\nIt is given that
\nStudents in the Art Club are designing a flag with the school’s mascot and emblem. The flag has four sides, with two sides that are twice as long as the other two sides.
\nNow,
\nWe know that,
\nA “Quadrilateral” is only a geometrical figure which has 4 sides
\nNow,
\nLet the length of the two sides be x cm
\nLet the length of the other two sides be 2x cm
\nWhere,
\nx = 1, 2, 3,……….., n
\nNow,
\nLet the value of x be 2
\nHence,
\nThe representations of the flag are:
\n\"\"<\/p>\n

Make Sense and Persevere: Is there more than one shape that could represent the flag?
\nAnswer:
\nYes, there is more than one shape that could represent the flag
\nNow,
\nThe representations of the flag are:
\n\"\"
\nHence,
\nFrom the above representation,
\nThe different shapes of the flags are: Rectangle, Parallelogram, and Trapezium, and a Quadrilateral<\/p>\n

Focus on math practices
\nReasoning How did you decide what lengths to use for the four sides of the flag you drew? What lengths could the actual flag be, based on your drawing?
\nAnswer:
\nIt is given that
\nThe flag has four sides, with two sides that are twice as long as the other two sides
\nNow,
\nLet the length of the two sides be x cm
\nLet the length of the other two sides be 2x cm
\nWhere,
\nx = 1, 2, 3,……….., n
\nNow,
\nBased on the drawing that is present in part (a),
\nThe actual lengths of the drawing are: 2 inches, 2 inches, 4 inches, and 4 inches<\/p>\n

Essential Question<\/strong>
\nHow can a shape that meets given conditions be drawn?
\nAnswer:
\nUse the conditions that are given to draw a shape and based on that shape, determine the name of the shape and its dimensions<\/p>\n

Try It!<\/strong>
\nUse a ruler and protractor to draw a quadrilateral with two equal sides that meet at a right angle, and two nonadjacent angles of the same measure. What is the name of the quadrilateral you drew?
\nThe quadrilateral I drew is a ___________
\nAnswer:
\nIt is given that
\nUse a ruler and protractor to draw a quadrilateral with two equal sides that meet at a right angle, and two nonadjacent angles of the same measure.
\nNow,
\nThe steps to draw the quadrilateral are:
\nStep 1:
\nDraw the line segment AB with a certain length i.e., AB = 4 cm
\nStep 2:
\nFrom B, make an angle of 90\u00b0 with the same length as AB and draw a line segment BC
\nStep 3:
\nFrom C, make another angle of 90\u00b0 and draw a line segment CD
\nHence, from the above steps,
\nWe can conclude that the quadrilateral drawn is: Square
\n\"\"<\/p>\n

Convince Me! Could you have drawn more than one shape that fits the given conditions? Explain.
\nAnswer:
\nNo, we can’t draw any shape other than the square that fits the above conditions<\/p>\n

Try It!<\/strong>
\na. Make a sketch to show another way Mr. Miller can arrange the desks to seat 10 people for lunch.
\nAnswer:
\nThe representation of the arrangement of desks to seating 10 people for lunch is:
\n\"\"
\nHence, from the above arrangement,
\nWe can conclude that Mr.Millercan arrange 6 desks in 2 rows to seat 10 people for lunch<\/p>\n

b. Use geometry software to make a rhombus with a side length of 6 units and two angles that measure 45\u00b0.
\nAnswer:
\nTo draw a rhombus,
\nThe geometry software we can use is: Geogebra
\nNow,
\nTo draw a rhombus, the following steps have to be followed:
\nStep 1:
\nDraw a line segment AB with a length of 6 cm
\nStep 2:
\nFrom A, make an angle of 45\u00b0 and join that segment and name it as AD
\nStep 3:
\nFrom D, draw a line segment DC = 6 cm
\nStep 4:
\nFrom C, make an angle of 45\u00b0 and join A and C
\nHence, from the above steps,
\nThe representation of the rhombus is:
\n\"\"<\/p>\n

KEY CONCEPT<\/strong>
\nYou can draw shapes that meet given conditions freehand, with a ruler and protractor, or with technology, The given conditions may include properties of geometric figures and relationships between parts of the figures.<\/p>\n

Use Appropriate Tools Deciding how precise the drawing of the shape should be will help you choose the method you use to draw the shape.<\/p>\n

Do You Understand?<\/strong>
\nQuestion 1.
\nEssential Question How can a shape that meets given conditions be drawn?
\nAnswer:
\nUse the conditions that are given to draw a shape and based on that shape, determine the name of the shape and its dimensions<\/p>\n

Question 2.
\nUse Appropriate Tools How can you decide whether to draw a shape freehand, with a ruler and protractor, or using technology?
\nAnswer:
\na. We will draw a shape freehand when there are not any measurements to draw a particular shape but only the name of the shape
\nb. We will draw a shape with a ruler when the measurements are specified for a particular shape
\nc. We will draw a shape with a protractor and a ruler when the measurements and an angle for a particular shape are specified<\/p>\n

Question 3.
\nConstruct Arguments Why can you draw more than one quadrilateral using four right angles?
\nAnswer:
\nWe know that,
\nA parallelogram is a quadrilateral with 2 pairs of opposite sides parallel. A rectangle is a special parallelogram that has 4 right angles. A square is a special rectangle that has all four sides congruent. The angle between these two sides could be a right angle
\nHence, from the above,
\nWe can conclude that we can draw more than one quadrilateral using four right angles<\/p>\n

\"Involving<\/p>\n

Do You Know How?<\/strong>
\nQuestion 4.
\nDraw, freehand, a quadrilateral with exactly one pair of parallel sides and at least one angle measuring 45\u00b0.
\nAnswer:
\nThe steps to draw a freehand quadrilateral with exactly one pair of parallel sides and at least one angle measuring 45\u00b0 are:
\nStep 1:
\nDraw a line segment AB with a certain length
\nStep 2:
\nMark an angle of 45\u00b0 from point A using the protractor and join the segment and name it as BC
\nStep 3:
\nFrom B, draw a line segment such that the length of AB must not be equal to the line segment that is drawn from B and name it BC
\nStep 4 :
\nFrom C, draw a line segment named CDand the length of CD must be equal to the length of AB since they are 2 parallel sides
\nStep 5:
\nJoin D and A and so we will get the desired quadrilateral
\n\"\"<\/p>\n

Question 5.
\nUse a ruler and protractor to draw a quadrilateral with four right angles, two side lengths each measuring 3 inches, and two side lengths each measuring 4 inches. What is the most descriptive name of the figure you drew?
\nAnswer:
\nIt is given that
\nUse a ruler and protractor to draw a quadrilateral with four right angles, two side lengths each measuring 3 inches, and two side lengths each measuring 4 inches
\nSo,
\nBy using the Geometry software,
\nThe representation of the quadrilateral with the given specifications is:
\n\"\"
\nHence, from the above,
\nWe can conclude that the suitable name of the quadrilateral that was drawn using the given specifications is: Rectangle<\/p>\n

Question 6.
\nUse geometry software to draw a quadrilateral with two angles measuring 80\u00b0 and two angles measuring 100\u00b0. What is the name of the figure you drew?
\nAnswer:
\nIt is given that
\nUse geometry software to draw a quadrilateral with two angles measuring 80\u00b0 and two angles measuring 100\u00b0.
\nSo,
\nThe representation of the quadrilateral with the given specifications is:
\n\"\"
\nNow,
\nFrom the above figure,
\nWe can observe that
\na. All the angles are not equal to 90\u00b0
\nb. All the side lengths are equal
\nHence, from the above,
\nWe can conclude that the quadrilateral we were drawn using the given specifications is: Rhombus<\/p>\n

Practice & Problem Solving<\/strong><\/p>\n

Question 7.
\nWhat quadrilaterals can you draw that have exactly four right angles?
\nAnswer:
\nThe quadrilaterals that we can draw that have exactly four right angles are:
\na. Rectangle b. Square<\/p>\n

Question 8.
\nA four-sided sandbox has more than two right angles, two side lengths of 2 feet, and two side lengths of 5 feet. What geometric shape best describes the shape of the sandbox?
\nAnswer:
\nIt is given that
\nA four-sided sandbox has more than two right angles, two side lengths of 2 feet, and two side lengths of 5 feet
\nSo,
\nThe representation of the sandbox with the given specifications is:
\n\"\"
\nNow,
\nFrom the above figure,
\nWe can observe that
\na. All the angle measures are 90\u00b0
\nb. The parallel sides are of the same length
\nHence, from the above,
\nWe can conclude that the shape of the given sandbox is: Rectangle<\/p>\n

Question 9.
\nWhat quadrilateral can you draw that has exactly one pair of parallel sides?
\nAnswer:
\nThe quadrilateral you can draw that has exactly one pair of parallel sides is: Trapezoid<\/p>\n

Question 10.
\nA friend is building a 4-sided garden with two side lengths of 19 feet and exactly one right angle. What quadrilaterals could describe the garden?
\nAnswer:
\nIt is given that
\nA friend is building a 4-sided garden with two side lengths of 19 feet and exactly one right angle.
\nSo,
\nFrom the above specifications,
\nThe shape of the garden must be a rectangle
\nSo,
\nThe representation of the 4-sided garden with the given specifications is:
\n\"\"
\nHence, from the above,
\nWe can conclude that the quadrilateral that could describe the garden is: Rectangle<\/p>\n

Question 11.
\nWhat quadrilaterals can you draw that have two side lengths of 9 centimeters and two side lengths of 4 centimeters?
\nAnswer:
\nThe quadrilaterals that you can draw that have two side lengths of 9 cm and two side lengths of 4 cm are:
\na. Rectangle (With all the angle measures 90\u00b0)
\nb. Parallelogram ( WIth all the angle measures not equal to 90\u00b0)
\nHence,
\nThe representation of the two quadrilateral with the given specifications is:
\n\"\"<\/p>\n

Question 12.
\nA park has a pond shaped like a quadrilateral with side lengths of 17 feet and no right angles. What other geometric shapes could describe the shape of the pond?
\nAnswer:
\nIt is given that
\nA park has a pond shaped like a quadrilateral with side lengths of 17 feet and no right angles
\nNow,
\nWe know that,
\nA quadrilateral that has all the side lengths equal and no right angles is: Rhombus
\nSo,
\nThe shape of the pond is: Rhombus
\nNow,
\nWe know that,
\nAny other geometric shape does not have all the equal side lengths other than the square but in the square, all the angle measures are 90\u00b0
\nHence, from the above,
\nWe can conclude that the shape of the pond is: Rhombus
\n\"\"\u00a0 Scale: 1 cm = 2 feet<\/p>\n

Question 13.
\nDraw a quadrilateral that has one angle measure of 20\u00b0 and exactly one side length of 4 units.
\nAnswer:
\nIt is given that to draw a quadrilateral that has one angle measure of 20\u00b0 and exactly one side length of 4 units.
\nHence,
\nThe representation of the quadrilateral with the given specifications by using the geometry software is:
\n\"\"<\/p>\n

Question 14.
\nWhich of the following shapes are trapezoids that have side lengths of 7 inches and 5 inches and a right angle? Select all that apply.
\n\"Envision
\nAnswer:
\nWe know that,
\nA right-angled Trapezoid has 2 adjacent right angles and a pair of parallel sides
\nHence,
\nThe Trapezoids that have represented by the given specifications are:
\n\"\"<\/p>\n

Question 15.
\nUsing computer software, draw a quadrilateral with two sets of parallel sides and two angles measuring 135 degrees.
\nAnswer:
\nIt is given that
\nDraw a quadrilateral with two sets of parallel sides and two angles measuring 135 degrees.
\nHence,
\nThe representation of the quadrilateral with the given specifications by using the geometry software is:
\n\"\"<\/p>\n

Question 16.
\nHigher-Order Thinking<\/strong> Draw a rhombus with side lengths of 6 units and angle measures of 100\u00b0, 80\u00b0, 100\u00b0, and 80\u00b0.
\nAnswer:
\nIt is given that
\nDraw a rhombus with side lengths of 6 units and angle measures of 100\u00b0, 80\u00b0, 100\u00b0, and 80\u00b0.
\nHence,
\nThe representation of a rhombus with the given specifications by using the geometry software is:
\n\"\"<\/p>\n

Assessment Practice<\/strong>
\nQuestion 17.
\nThomas is painting a geometry mural. He is painting quadrilaterals that have at least 1 line of symmetry.
\nPART A
\nWhich could be a quadrilateral that Thomas painted? Select all that apply.
\n\"Envision
\nAnswer:
\nIt is given that
\nThomas is painting a geometry mural. He is painting quadrilaterals that have at least 1 line of symmetry.
\nNow,
\nWe know that,
\nThe “Symmetry lines” are the lines that have equal side lengths
\nHence, from the above,
\nThe quadrilaterals that Thomas painted are:
\n\"\"<\/p>\n

PART B
\nWhich of the following figures can he also include in his painting?
\nA. quadrilateral with no equal sides
\nB. A quadrilateral with only 2 equal sides that are perpendicular to each other
\nC. A quadrilateral with 2 pairs of equal sides and 1 right angle
\nD. A quadrilateral with 2 pairs of parallel equal sides, with no right angles
\nAnswer:
\nFrom part (a),
\nWe can observe that the quadrilaterals that are included in Thomas painting have:
\na. The equal sides and 1 right angle (or) more than one right angle
\nb. The equal pair of sides and no right angles
\nHence, from the above,
\nWe can conclude that the nature of figures that are included in Thomas paintings are: Option C and Option\u00a0 D<\/p>\n

Lesson 8.3 Draw Triangles with Given Conditions<\/h3>\n

Solve & Discuss It!<\/strong><\/p>\n

Kane has 4 pieces of wood available to build a triangle-shaped garden. Which pieces of wood can he use?
\n\"Envision
\nI can… draw triangles when given information about their side lengths and angle measures.
\nAnswer:
\nIt is given that
\nKane has 4 pieces of wood available to build a triangle-shaped garden.
\nNow,
\nWe know that,
\nTo build a Triangle,
\nThe sum of the lengths of the two shortest sides must be greater than the length of the third side
\nNow,
\nThe given pieces of wood are:
\na. 2 feet b, 3 feet c. 4 feet d. 5 feet
\nSo,
\nThe possible combination of pieces of wood is:
\na. (2, 3, 5) b. (2, 3, 4) c. (3, 4, 5) d. (2, 4, 5)
\nNow,
\nThe possible combination of pieces of wood that allow to form a triangle-shaped garden is:
\na. 2 + 3 !>[Not greater thhan] 5
\nb. 2 + 3 > 4
\nc. 3 + 4 > 5
\nd. 2 + 4 > 5
\nHence, from the above,
\nWe can conclude that the possible combinations of the pieces of wood that form a Triangle-shaped garden are:
\na. (2 feet, 3 feet, 4 feet)
\nb. (3 feet, 4 feet, 5 feet)
\nc. (2 feet, 4 feet, 5 feet)<\/p>\n

Make Sense and Persevere Try all possible combinations of three pieces of wood.
\nAnswer:
\nWe know that,
\nTo build a Triangle,
\nThe sum of the lengths of the two shortest sides must be greater than the length of the third side
\nNow,
\nThe given pieces of wood from part (a) are:
\na. 2 feet b, 3 feet c. 4 feet d. 5 feet
\nSo,
\nThe possible combination of pieces of wood to form a triangle-shaped garden is:
\na. (2, 3, 5) b. (2, 3, 4) c. (3, 4, 5) d. (2, 4, 5)
\nNow,
\nThe possible combination of pieces of wood that allow to form a triangle-shaped garden is:
\na. 2 + 3 !>[Not greater thhan] 5
\nb. 2 + 3 > 4
\nc. 3 + 4 > 5
\nd. 2 + 4 > 5
\nHence, from the above,
\nWe can conclude that the possible combinations of the pieces of wood that form a Triangle-shaped garden are:
\na. (2 feet, 3 feet, 4 feet)
\nb. (3 feet, 4 feet, 5 feet)
\nc. (2 feet, 4 feet, 5 feet)<\/p>\n

Focus on math practices
\nUse Structure Are there any combinations of three pieces of wood that will not create a triangle? Explain.
\nAnswer:
\nYes, there is a combination of three pieces of wood that will not create a triangle
\nNow,
\nFrom part (b),
\nWe can observe that the combination of three pieces of wood that will not create a triangle is: (2, 3, 5)
\nHence, from the above,
\nWe can conclude that
\nThe combination of three pieces of wood that will not create a triangle is: (2 feet, 3 feet, 5 feet)<\/p>\n

\"Involving<\/p>\n

Essential Question<\/strong>
\nHow can you determine when it is possible to draw a triangle given certain conditions?
\nAnswer:
\nThe condition to draw a Triangle under the given conditions is:
\nThe sum of the lengths of the two shortest sides must be greater than the length of the third side<\/p>\n

Try It!<\/strong>
\nHow many unique triangles can be drawn with given side lengths of 8 inches, 10.3 inches, and 13 inches?
\n_________ unique triangle(s) can be drawn with the given side lengths.
\nAnswer:
\nThe given side lengths are: 8 inches, 10.3 inches, and 13 inches
\nNow,
\nThe representations of the given side lengths of the triangle in different positions are:
\n\"\"
\nSo,
\nFrom the above,
\nWe can observe that the triangles with the same side lengths have the same shape and the same size, no matter how they are positioned
\nHence, from the above,
\nWe can conclude that “1 unique Triangle” can be drawn with the given side lengths<\/p>\n

Convince Me! When two sides of a triangle are switched, why is it still considered the same triangle?
\nAnswer:
\nWhen two sides of a triangle are switched, we are only switching their positions with their original side lengths.
\nSo,
\nIt is still considered the same triangle<\/p>\n

Try It!<\/strong>
\na. Write three side lengths that will form a triangle. Write three side lengths that will NOT form a triangle.
\nAnswer:
\nWe know that,
\nIn order to construct a Triangle,
\nThe sum of the lengths of the two shortest sides must be greater than the length of the third side
\nHence, from the above,
\nWe can conclude that
\nThe three side lengths that form a triangle are: 3 units, 4 units, and 5 units
\nThe three side lengths that will not form a triangle are: 2 units, 3 units, and 5 units<\/p>\n

b. Can a triangle be drawn with a side length of 3 inches and angles at each end measuring 90\u00b0 and 89\u00b0? Explain.
\nAnswer:
\nIt is given that
\nThe side length of a triangle is: 3 inches
\nThe angle measures of a triangle are 90\u00b0 and 89\u00b0
\nNow,
\nWe know that,
\nA triangle can only have 1 right angle
\nBut,
\nThe given triangle has 1 right angle and another almost right angle with a third angle measure 1\u00b0
\nNow,
\nThe representation of the triangle with the given specifications will be like:
\n\"\"
\nHence, from the above,
\nWe can conclude that
\nIt is not possible to draw a triangle with a side length of 3 inches and angles at each end measuring 90\u00b0 and 89\u00b0<\/p>\n

Try It!<\/strong>
\nCan more than one triangle be drawn with two side lengths of 6 inches and a non-included angle of 60\u00b0? Explain.
\nAnswer:
\nIt is given that
\nThe two side lengths of the triangle are: 6 inches and 6 inches
\nThe non-included angle is: 60\u00b0
\nNow,
\nWe know that,
\nAs the two side lengths are the same, the triangle will be an “Isosceles Triangle”
\nSo,
\nBesides the given angle measure, the other two angle measures must be the same
\nSo,
\nThe other two angle measures will be: 60\u00b0 and 60\u00b0 [ Since the first angle measure is 60\u00b0 and the remaining angle measure is 120\u00b0]
\nSo,
\nThe representation of the triangle with the given specifications is:
\n\"\"
\nNow,
\nFrom the above triangles,
\nWe can observe that there is no change in shape irrespective of the same side lengths and the same non-included angle
\nHence, from the above,
\nWe can conclude that more than one triangle can not be drawn with two side lengths of 6 inches and a non-included angle of 60\u00b0<\/p>\n

KEY CONCEPT<\/strong>
\nYou can analyze given conditions of side lengths and angle measures to determine whether one unique triangle, more than one unique triangle, or no triangle can be drawn.
\nThere is more than one possible triangle given these cases: all three angles, or two sides and a nonincluded angle.
\n\"Envision
\nThere is one unique triangle given these cases: all three sides, two sides and an included angle, or two angles and an included side.
\n\"Envision<\/p>\n

Do You Understand?<\/strong>
\nQuestion 1.
\nEssential Question How can you determine when it is possible to draw a triangle given certain conditions?
\nAnswer:
\nThe condition to draw a Triangle under the given conditions is:
\nThe sum of the lengths of the two shortest sides must be greater than the length of the third side<\/p>\n

Question 2.
\nLook for Relationships What is the relationship between all triangles that can be drawn given the same three angle measures?
\nAnswer:
\nIf we have all the same three angle measures, then that triangle is known as an “Equilateral triangle”
\nSo,
\nWhatever the turns, reflection, etc., we did, the shape of the equilateral triangle will not change
\nHence, from the above,
\nWe can conclude that a “Unique Triangle” can be formed if we draw a triangle will the same three angle measures<\/p>\n

Question 3.
\nWhy can there be only one way to draw a triangle if two sides and an included angle are given?
\nAnswer:
\nIt is possible to draw more than one triangle has the side lengths and angle measure as given. It depends on which line you start with, which end of the line you draw the angles, and whether they are above or below the line or not.
\nHence, from the above,
\nWe can conclude that because of the above, there is only one way to draw a triangle if two sides and an included angle are given<\/p>\n

Do You Know How?<\/strong>
\nQuestion 4.
\nHow many triangles can be drawn with side lengths 4 centimeters, 4.5 centimeters, and 9 centimeters? Explain.
\nAnswer:
\nThe given side lengths of a triangle are: 4 cm, 4.5 cm, and 9 cm
\nNow,
\nWe know that,
\nIn order to construct a triangle,
\nThe sum of the length of the two shortest sides must be greater than the length of the third side
\nSo,
\n4 + 4.5 > 9
\n8.5 >!\u00a09
\nHence, from the above,
\nWe can conclude that the triangles can not be drawn for the given side lengths since the condition to form a triangle is not true for the given side lengths<\/p>\n

Question 5.
\nCan more than one triangle be drawn with side lengths of 5 inches and 7 inches and an included angle with a measure of 50\u00b0? Explain.
\nAnswer:
\nThe given side lengths of a triangle are: 5 inches and 7 inches
\nThe included angle of a triangle is: 50\u00b0
\nNow,
\nThe representations of the triangles with the given specifications are:
\n\"\"
\nNow,
\nFrom the above,
\nWe can observe that the triangle with the given specifications is the same even we turn the triangle to any degree
\nHence, from the above,
\nWe can conclude that
\nWe can not draw more than one triangle with side lengths of 5 inches and 7 inches and an included angle with a measure of 50\u00b0<\/p>\n

Question 6.
\nSketch two different triangles that have angle measures of 45\u00b0, 45\u00b0, and 90\u00b0.
\nAnswer:
\nThe representation of the different triangles that have angle measures of 45\u00b0, 45\u00b0, and 90\u00b0 are:
\n\"\"<\/p>\n

Practice & Problem Solving<\/strong><\/p>\n

Question 7.
\nDraw two different triangles with angle measurements of 90\u00b0, 35\u00b0, and 55\u00b0.
\nAnswer:
\nThe representation of the two different triangles with angle measurements of 90\u00b0, 35\u00b0, and 55\u00b0 is:
\n\"\"<\/p>\n

Question 8.
\nIf you form a triangle from three given side lengths, will you always get one triangle or more than one triangle?
\nAnswer:
\nIf you form a triangle from three given side lengths, then you will not always get only one triangle
\nEx:
\nThe given side lengths of a triangle are: 3 cm, 4 cm, and 5 cm
\nNow,
\nWe know that,
\nIn order to construct a triangle,
\nThe sum of the lengths of the two short sides must be greater than the length of the third side
\nSo,
\nThe possible combinations for forming a triangle are: (3, 4, 5), (3, 5, 4), (4, 3, 5), (5, 3, 4)<\/p>\n

Question 9.
\nHow can you make different-looking triangles given two of the angle measures and the included side lengths?
\nAnswer:
\nSince the two angles are already given, therefore the value of the third angle is already fixed. That is, the third angle is the difference between 180\u00b0 and the sum of the two angles. If the angles and the side lengths are given, then we can form any number of angles as long as the ratio of angles are consistent<\/p>\n

Question 10.
\nIf you form a triangle from two given angle measures and the length of the included side, will you always get one triangle or will you get more than one triangle?
\nAnswer:
\nIt is not always possible to construct a triangle from a given side length and two angles. If the two given angles add to more than 180\u00b0, then the sides of the triangle will diverge and never meet.<\/p>\n

Question 11.
\nHow can you make different triangles with the same angle measures?
\nAnswer:
\nWe can make different triangles with the same angle measures by making the sum of the angles consistent
\nExample:
\nThe given angle measures are: 30\u00b0, 70\u00b0, 80\u00b0
\nNow,
\nThe combination of different angle measures to make different triangles are:
\na. 30\u00b0, 60\u00b0 90\u00b0
\nb. 40\u00b0, 50\u00b0, 90\u00b0
\nc. 40\u00b0, 60\u00b0, 80\u00b0<\/p>\n

Question 12.
\nGiven two side lengths of 15 units and 9.5 units, with a nonincluded angle of 75\u00b0, can you draw no triangles, only one triangle, or more than one triangle?
\nAnswer:
\nThe given side lengths are: 15 units and 9.5 units
\nThe non-included angle is: 75\u00b0
\nNow,
\nThe representation of the triangle with the given specifications are:
\n\"\"
\nNow,
\nFrom the above,
\nWe can observe that there is more than one triangle with the given specifications
\nHence, from the above,
\nWe can conclude that we can draw more than one triangle with the given specifications<\/p>\n

Question 13.
\nA student was asked to form different triangles with angle measures of 90\u00b0, 30\u00b0, and 60\u00b0. She incorrectly said this triangle is the only triangle with angle measures of 90\u00b0, 30\u00b0, and 60\u00b0. What mistake might she have made?
\n\"Envision
\nAnswer:
\nIt is given that
\nA student was asked to form different triangles with angle measures of 90\u00b0, 30\u00b0, and 60\u00b0. She incorrectly said this triangle is the only triangle with angle measures of 90\u00b0, 30\u00b0, and 60\u00b0
\nNow,
\nWe know that,
\nWe can form any number of triangles as long as the sum of the angles is 180\u00b0
\nSo,
\nThe other combinations of the given angle measures are:
\na. 30\u00b0, 60\u00b0, 90\u00b0
\nb. 40\u00b0, 60\u00b0, 80\u00b0
\nc. 30\u00b0, 50\u00b0, 100\u00b0
\nHence, from the above,
\nWe can conclude that the mistake done by her is:
\nShe did not consider the other shapes of the triangles by adjusting the given angle measures<\/p>\n

Question 14.
\nIn triangle QRS, m\u2220QSR = 100\u00b0, m\u2220SQR = 45\u00b0, and QR = 4 units. In triangle XYZ, m\u2220XYZ = 100\u00b0, m\u2220ZXY = 45\u00b0, and XY = 4 units. Are triangles QRS and XYZ the same? Explain.
\nAnswer:<\/p>\n

Question 15.
\nYou are asked to make a triangular sign using the given information about triangle WXY. In triangle WXY, m\u2220WXY = 45\u00b0, m\u2220YWX = 90\u00b0, and WX = 5 feet.
\na. Which triangle is correct? Each square on the grid is equal to 1 square foot.
\n\"Envision
\nAnswer:
\nIt is given that
\nYou are asked to make a triangular sign using the given information about triangle WXY. In triangle WXY, m\u2220WXY = 45\u00b0, m\u2220YWX = 90\u00b0, and WX = 5 feet.
\nSo,
\nThe representation of the triangle with the given specifications is:
\n\"\"
\nHence, from the above,
\nWe can conclude that Option B matches the given above figure<\/p>\n

b. Explain why only one triangle can be formed with these three pieces of information.
\nAnswer:
\nWe know that,
\nIt is not always possible to construct a triangle from a given side length and two angles. If the two given angles add to more than 180\u00b0, then the sides of the triangle will diverge and never meet.
\nHence, from the above,
\nWe can conclude that only one triangle can be formed by the given information in part (a)<\/p>\n

Question 16.
\nLook for Relationships Two different triangles have side lengths of 13 and 16 units and a non-included angle of 50\u00b0. Explain how the triangles are different.
\nAnswer:
\nIt is given that
\nTwo different triangles have side lengths of 13 and 16 units and a non-included angle of 50\u00b0
\nNow,
\nThe representations of the triangle with the given specifications are: (Scale: 1 cm = 2 units)
\n\"\"
\nNow,
\nFrom the above,
\nWe can observe that the angles are different for the two representations
\nHence, from the above,
\nWe can conclude that the representations of the triangles are different due to the remaining angle measures other than the non-included angle measure<\/p>\n

Question 17.
\nHigher-Order Thinking Two triangles have side lengths of 12 units and 15 units and a non-included angle of 45\u00b0. Draw two different triangles with these conditions.
\nAnswer:
\nIt is given that
\nTwo triangles have side lengths of 12 units and 15 units and a non-included angle of 45\u00b0
\nHence,
\nThe representation of the triangle with the given specifications are:
\n\"\"<\/p>\n

Question 18.
\nFor triangle RST, RS is 12 centimeters, ST is 16 centimeters, and RT is 19 centimeters. How many triangles can be drawn with the given side lengths?
\nAnswer:
\nIt is given that
\nFor triangle RST, RS is 12 centimeters, ST is 16 centimeters, and RT is 19 centimeters.
\nSo,
\nThe representation of the triangle with the given specifications is:
\n\"\"
\nHence, from the above,
\nWe can conclude that only one triangle can be drawn with the given side lengths<\/p>\n

Question 19.
\nA triangle has two side lengths of 8.5 centimeters and 9.5 centimeters. What is a possible length for the third side? Explain why this is a possible length.
\nAnswer:
\nIt is given that
\nA triangle has two side lengths of 8.5 centimeters and 9.5 centimeters.
\nNow,
\nWe know that,
\nIn order to construct a triangle,
\nThe sum of the lengths of the two shortest sides must be greater than the length of the longer side
\nNow,
\nLet the third side be x
\nCase 1:
\n8.5 + 9.5 > x
\n18 > x
\nx < 18
\nCase 2:
\n8.5 + x > 9.5
\nx > 9.5 – 8.5
\nx > 1
\nHence, from the above,
\nWe can conclude that the possible length of the third side must be: 1 < x < 18<\/p>\n

Question 20.
\nCan a triangle be formed with side lengths of 4, 5, and 7 units?
\nAnswer:
\nThe given side lengths are: 4 units, 5 units, and 7 units
\nNow,
\nWe know that,
\nIn order to construct a triangle,
\nThe sum of the lengths of the two shortest sides must be greater than the length of the longer side
\nSo,
\n4 + 5 > 7
\n9 > 7
\nHence, from the above,
\nWe can conclude that the triangle can be formed wit the given side lengths 4, 5, and 7 units<\/p>\n

Assessment Practice<\/strong>
\nQuestion 21.
\nWhich of the following combinations of side lengths would form a triangle? Select all that apply.
\n\u2610 7 in., 10 in., 2.5 in.
\n\u2610 4.5 ft, 8 ft, 5 ft
\n\u2610 5 yd, 11 yds, 5 yd
\n\u2610 12 in., 5 in., 9.5 in.
\n\u2610 7 m, 7 m, 9 m
\n\u2610 6 ft, 16 ft, 9 ft
\nAnswer:
\nWe know that,
\nIn order to construct a triangle,
\nThe sum of the lengths of the two shortest sides must be greater than the length of the longer side
\nHence,
\nThe combinations of side lengths that would form a triangle are:
\n\"\"<\/p>\n

Question 22.
\nWhich of the following combinations of side lengths would NOT form a triangle?
\nA. 7 cm, 10 cm, 13 cm
\nB. 10 ft, 13 ft, 15 ft
\nC. 10 yd, 11 yd, 13 yd
\nD. 10 in., 13 in., 23 in.
\nAnswer:
\nWe know that,
\nIn order to construct a triangle,
\nThe sum of the lengths of the two shortest sides must be greater than the length of the longer side
\nHence,
\nThe combinations of side lengths that would not form a triangle are:
\n\"\"<\/p>\n

Question 23.
\nDraw a triangle that has exactly one line of symmetry.
\n\"Envision
\nAnswer:
\nWe know that,
\nThe triangle, which has two equal sides, two equal angles and exactly one line of symmetry is known as the Isosceles Triangle. So, \u0394ABC is an Isosceles Triangle. Therefore, the triangle which has only one line of symmetry has been known as the Isosceles Triangle.
\nHence,
\nThe representation of the triangle that has exactly on the line of symmetry is:
\n\"\"<\/p>\n

Lesson 8.4 Solve Problems Using Angle Relationships<\/h3>\n

Explore It!<\/strong><\/p>\n

The intersecting skis form four angles.
\n\"Envision<\/p>\n

I can… solve problems involving angle relationships.<\/p>\n

A. List all the pairs of angles that share a ray.
\nAnswer:
\nIt is given that the intersecting skis form four angles.
\nNow,
\nThe representation of the intersecting skies and the angles made by intersection are:
\n\"Envision
\nHence, from the above figure,
\nWe can conclude that the angles that share a ray are:
\n\u22201, \u22202, \u22203, and \u22204<\/p>\n

B. Suppose the measure of Z1 increases. What happens to the size of \u22202? \u22203?
\nAnswer:
\nFrom the given figure,
\nWe can observe that \u22201 and \u22202 are on the same side and they are known as “Adjacent angles”
\nWe can observe that \u22201 and \u22203 are on the opposite sides of the ray and they are known as “Vertical angles”
\nNow,
\nWe know that,
\nThe sum of the adjacent angles is 180\u00b0
\nThe angle measures of the vertical angles are the same
\nHence, from the above,
\nWe can conclude that
\nWhen the value of \u22201 increases, the value of \u22202 automatically decreases
\nWhen the value of \u22201 increases, the value of \u22203 also increases<\/p>\n

C. How does the sum of the measures of \u22201 and \u22202 change when one ski moves? Explain.
\nAnswer:
\nFrom the given figure,
\nWe can observe that \u22201 and \u22202 lies on the same side of the ski and they are known as “Adjacent angles” (or) “Supplementary angles”
\nNow,
\nWe know that,
\nThe sum of the angle measures of adjacent angles is 180\u00b0
\nHence, from the above,
\nWe can conclude that the sum of the measures of \u22201 and \u22202 always remains constant even when one ski moves<\/p>\n

Focus on math practices<\/strong>
\nConstruct Arguments Why does the sum of all four angle measures stay the same when one of the skis moves?
\nAnswer:
\nFrom the given figure,
\nWe can observe that the intersecting skies form 4 angles
\nNow,
\nWe know that,
\nThe sum of the adjacent angles’ angle measures is 180\u00b0
\nThe angle measures of the vertical angles are the same
\nSo,
\n\u22201 + \u22202 = 180\u00b0 and \u22203 + \u22204 = 180\u00b0
\n\u22201 = \u22203 and \u22202 = \u22204
\nSo,
\n\u22201 + \u22202 + \u22203 + \u22204 = 360\u00b0
\nSo,
\nEven when we change one angle measure, the remaining angle measures will adjust their angle measures accordingly so that the sum of all the four angles remain the same i.e, 360\u00b0
\nHence, from the above,
\nWe can conclude that the sum of all four angle measures stay the same even when one of the skis moves<\/p>\n

Essential Question<\/strong>
\nHow are angles formed by intersecting lines related?
\nAnswer:
\nWhen two lines intersect,\u00a0the angles that are opposite each other are vertical angles. Recall that a linear pair is a pair of adjacent angles whose non-common sides are opposite rays. So, when two lines intersect, the angles that are on the same side of a line form a linear pair.<\/p>\n

Try It!<\/strong>
\n\u2220MNQ and \u2220PNR are vertical angles. What is the value of x?
\n\"Envision
\nVertical angles are __________, so the equation ___________ can be used to find x. The value of x is __________.
\nAnswer:
\nIt is given that \u2220MNQ and \u2220PNR are vertical angles
\nNow,
\nThe given figure is:
\n\"Envision
\nNow,
\nWe know that,
\nThe angle measures of the vertical angles are the same
\nSo,
\n(3x – 6)\u00b0 = 114\u00b0
\n3x\u00b0 = 114\u00b0 + 6\u00b0
\n3x\u00b0 = 120\u00b0
\nx\u00b0 = \\(\\frac{120\u00b0}{3}\\)
\nx\u00b0 = 40\u00b0
\nHence, from the above,
\nWe can conclude that
\nSince the vertical angles are congruent, the equation (3x – 6)\u00b0 = 114\u00b0 can be used to find the value of x
\nHence,
\nThe value of x is: 40\u00b0<\/p>\n

Convince Me! Why can you use an equation when solving for x in the diagram?
\nAnswer:
\nThe given figure is:
\n\"Envision
\nNow,
\nWe know that,
\nThe vertical angles are always congruent
\nSo,
\nThe angle measures of the vertical angles are also congruent
\nNow,
\nFrom the given figure,
\nWe can observe that \u2220MNQ and \u2220PNR are vertical angles
\nSo,
\n\u2220MNQ = \u2220PNR
\nSo,
\n(3x – 6)\u00b0 = 114\u00b0
\nHence, from the above,
\nWe can conclude that we will use the equation to solve for x since the given angles are the vertical angles<\/p>\n

Try It!
\nm\u22201 is 4 times m\u22202. \u22201 and \u22202 are complementary. \u22201 and \u22203 are vertical angles. \u22203 and \u22204 are supplementary. What are the measures of the four angles?
\nAnswer:
\nIt is given that
\nm\u22201 is 4 times m\u22202. \u22201 and \u22202 are complementary. \u22201 and \u22203 are vertical angles. \u22203 and \u22204 are supplementary
\nNow,
\nThe given relation between \u22201 and \u22202 is:
\n\u22202 = 4\u22201
\nNow,
\nFrom the given information,
\n\u22201 + \u22202 = 90\u00b0
\n\u22201 + 4\u22201 = 90\u00b0
\n5\u22201 = 90\u00b0
\n\u22201 = \\(\\frac{90\u00b0}{5}\\)
\n\u22201 = 18\u00b0
\nSo,
\n\u22202 = 4 \u22201
\n\u22202 = 4 (18\u00b0)
\n\u22202 = 72\u00b0
\nNow,
\nFrom the given information,
\n\u22201 and \u22203 are vertical angles. \u22203 and \u22204 are supplementary
\nSo,
\n\u22201 = \u22203 = 18\u00b0
\nNow,
\n\u22203 + \u22204 = 180\u00b0
\n\u22204 = 180\u00b0 – 18\u00b0
\n\u22204 = 162\u00b0
\nHence, from the above,
\nWe can conclude that the measures of the four angles are:
\n\u22201 = 18\u00b0, \u22202 = 72\u00b0, \u22203 = 18\u00b0, and \u22204 = 162\u00b0<\/p>\n

KEY CONCEPT<\/strong>
\n\"Envision<\/p>\n

Do You Understand?<\/strong><\/p>\n

Question 1.
\nEssential Question How are angles formed by intersecting lines related?
\nAnswer:
\nWhen two lines intersect,\u00a0the angles that are opposite each other are vertical angles. Recall that a linear pair is a pair of adjacent angles whose non-common sides are opposite rays. So, when two lines intersect, the angles that are on the same side of a line form a linear pair.<\/p>\n

Question 2.
\nUse Structure Can vertical angles also be adjacent angles? Explain.
\nAnswer:
\nNo, vertical angles can never be adjacent. Adjacent angles are the ones next to each other while vertical angles are opposite from each other.<\/p>\n

Question 3.
\nReasoning Do complementary and supplementary angles also have to be adjacent angles? Explain.
\nAnswer:
\nSupplementary and\u00a0<\/span>complementary angles do not have to be adjacent, but they can be possible. Supplementary angles are two angles whose measures sum to 180 \u00b0 and complementary are the sum have to add up to 90 \u00b0<\/span><\/p>\n

Do You Know How?<\/strong><\/p>\n

Use the diagram below for 4-6.
\n\"Envision
\nAnswer:
\nThe given figure is:
\n\"Envision<\/p>\n

Question 4.
\nList two pairs of adjacent angles.
\nAnswer:
\nFrom the given figure,
\nWe can observe that
\nThe two pairs of adjacent angles are:
\na. \u22201 and \u22202\u00a0 \u00a0 \u00a0 b. \u22203 and \u22204<\/p>\n

Question 5.
\nList all pairs of vertical angles.
\nAnswer:
\nFrom the given figure,
\nWe can observe that
\nThe pairs of vertical angles are:
\n\u22201 and \u22203<\/p>\n

Question 6.
\nIf \u22201 and \u22203 are the same measure, what is the value of x?
\nAnswer:
\nFrom the given figure,
\nWe can observe that
\n\u22201 = 9x\u00b0 and \u22203 = 90\u00b0
\nNow,
\nIt is given that \u22201 and \u22203 are the same measure
\nSo,
\n\u22201 = \u22203
\nSo,
\n9x\u00b0 = 90\u00b0
\nx = \\(\\frac{90\u00b0}{9}\\)
\nx = 10\u00b0
\nHence, from the above,
\nWe can conclude that the value of x is: 10\u00b0<\/p>\n

Practice & Problem Solving<\/strong><\/p>\n

Question 7.
\nList each angle adjacent to \u2220w.
\n\"Envision
\nAnswer:
\nThe given figure is:
\n\"Envision
\nNow,
\nFrom the given figure,
\nWe can observe that
\nThe angles adjacent to \u2220w are: \u2220x and \u2220z
\nHence, from the above,
\nWe can conclude that the angles adjacent to \u2220w are: \u2220x and \u2220z<\/p>\n

Question 8.
\nList two pairs of adjacent angles.
\n\"Envision
\nAnswer:
\nThe given figure is:
\n\"Envision
\nNow,
\nWe know that,
\nThe “Adjacent angles” are the angles that share the common vertex and common side
\nNow,
\nFrom the given figure,
\nWe can observe that
\nThe adjacent angles are: \u2220KOL and \u2220NOM
\nHence, from the above,
\nWe can conclude that the two pairs of adjacent angles are: \u2220KOL and \u2220NOM<\/p>\n

Question 9.
\nFind the value of x.
\n\"Envision
\nAnswer:
\nThe given figure is:
\n\"Envision
\nNow,
\nFrom the given figure,
\nWe can observe that 9x\u00b0 and 18\u00b0 are the complementary angles
\nNow,
\nWe know that,
\nThe sum of the complementary angles is always 90\u00b0
\nSo,
\n9x\u00b0 + 18\u00b0 = 90\u00b0
\n9x\u00b0= 90\u00b0 – 18\u00b0
\n9x\u00b0 = 72\u00b0
\nx = \\(\\frac{72\u00b0}{9}\\)
\nx = 8\u00b0
\nHence, from the above,
\nWe can conclude that the value of x is: 8\u00b0<\/p>\n

Question 10.
\nFind the value of x.
\n\"Envision
\nAnswer:
\nThe given figure is:
\n\"Envision
\nNow,
\nFro the given figure,
\nWe can observe that 125\u00b0 and (5x + 30)\u00b0 are vertical angles
\nNow,
\nWe know that,
\nThe vertical angles are always congruent
\nSo,
\n(5x + 30)\u00b0 = 125\u00b0
\n5x\u00b0 = 125\u00b0 – 30\u00b0
\n5x\u00b0 = 95\u00b0
\nx = \\(\\frac{95\u00b0}{5}\\)
\nx\u00b0 = 19\u00b0
\nHence, from the above,
\nWe can conclude that the value of x is: 19\u00b0<\/p>\n

Question 11.
\n\u22201 and \u22202 are complementary angles. The measure of \u22201 is 42\u00b0. The measure of \u22202 is (3x)\u00b0. Find the value of x.
\n\"Envision
\nAnswer:
\nIt is given that
\n\u22201 and \u22202 are complementary angles. The measure of \u22201 is 42\u00b0. The measure of \u22202 is (3x)\u00b0.
\nNow,
\nThe given figure is:
\n\"Envision
\nNow,
\nWe know that,
\nThe sum of the complementary angles is always 90\u00b0
\nSo,
\n\u22201 + \u22202 = 90\u00b0
\n42\u00b0+ 3x\u00b0 = 90\u00b0
\n3x\u00b0 = 90\u00b0 – 42\u00b0
\n3x\u00b0 = 48\u00b0
\nx\u00b0 = \\(\\frac{48\u00b0}{3}\\)
\nx\u00b0= 16\u00b0
\nHence, from the above,
\nWe can conclude that the value of x is: 16\u00b0<\/p>\n

Question 12.
\nTwo streets form an intersection. \u2220C and \u2220D are supplementary angles. If the measure of \u2220C is 128\u00b0 and the measure of \u2220D is two times the value of x, what is the value of x?
\n\"Envision
\nAnswer:
\nIt is given that
\nTwo streets form an intersection. \u2220C and \u2220D are supplementary angles. The measure of \u2220C is 128\u00b0 and the measure of \u2220D is two times the value of x
\nNow,
\nThe given figure is:
\n\"Envision
\nNow,
\nWe know that,
\nThe sum of the supplemenatry angles is always 180\u00b0
\nSo,
\n\u2220C + \u2220D = 180\u00b0
\n128\u00b0 + 2x\u00b0 = 180\u00b0
\n2x\u00b0 = 180\u00b0 – 128\u00b0
\n2x\u00b0 = 52\u00b0
\nx\u00b0 = \\(\\frac{52\u00b0}{2}\\)
\nx\u00b0 = 26\u00b0
\nHence, from the above,
\nWe can conclude that the value of x is: 26\u00b0<\/p>\n

Question 13.
\nIf A and B are supplementary angles and \u2220A is three times as large as \u2220B, find the measures of \u2220A and \u2220B.
\nAnswer:
\nIt is given that
\nA and B are supplementary angles and \u2220A is three times as large as \u2220B
\nNow,
\nLet the value of \u2220B be x\u00b0
\nSo,
\nThe value of \u2220A will be: 3x\u00b0
\nNow,
\nWe know that,
\nThe sum of the supplementary angles is always 180\u00b0
\nSo,
\n\u2220A + \u2220B = 180\u00b0
\nSo,
\n3x\u00b0 + x\u00b0 = 180\u00b0
\n4x\u00b0 = 180\u00b0
\nx\u00b0 = \\(\\frac{180\u00b0}{4}\\)
\nx\u00b0= 45\u00b0
\nSo,
\n\u2220B = 45\u00b0
\n\u2220A = 3x\u00b0 = 3 (45\u00b0) = 135\u00b0
\nHence, from the above,
\nWe can conclude that the angle measures of \u2220A and \u2220B are 135\u00b0 and 45\u00b0 respectively<\/p>\n

Question 14.
\nHigher-Order Thinking<\/strong> The measure of \u2220DBE is (0.1x \u2013 22)\u00b0 and the measure of \u2220CBE is (0.3x \u2013 54)\u00b0. Find the value of x.
\n\"Envision
\nAnswer:
\nIt is given that
\nThe measure of \u2220DBE is (0.1x \u2013 22)\u00b0 and the measure of \u2220CBE is (0.3x \u2013 54)\u00b0
\nNow,
\nThe given figure is:
\n\"Envision
\nNow,
\nFrom the given figure,
\nWe can observe that \u2220DBE and \u2220CBE are the complementary angles
\nNow,
\nWe know that,
\nThe sum of the complementary angles is always 90\u00b0
\nSo,
\n\u2220DBE + \u2220CBE = 90\u00b0
\n(0.1x – 22)\u00b0 + (0.3x – 54)\u00b0 = 90\u00b0
\n0.4x\u00b0 – 76\u00b0 = 90\u00b0
\n0.4x\u00b0= 90\u00b0 + 76\u00b0
\n0.4x\u00b0 = 166\u00b0
\n4x\u00b0 = 1,660\u00b0
\nx\u00b0 = \\(\\frac{1,660\u00b0}{4}\\)
\nx\u00b0 = 415\u00b0
\nHence, from the above,
\nWe can conclude that the value of x is: 415\u00b0<\/p>\n

Question 15.
\nReasoning \u22201 and an angle that measures 50\u00b0 are supplementary. Another angle that measures 50\u00b0 and \u22203 is supplementary. Show that m\u22201 and m\u22203 are equal.
\nAnswer:
\nIt is give that
\n\u22201 and an angle that measures 50\u00b0 are supplementary. Another angle that measures 50\u00b0 and \u22203 is supplementary
\nNow,
\nWe know that,
\nThe sum of the supplementary angles is always 180\u00b0
\nNow,
\nLet the unkown angle be \u2220x
\nNow,
\n\u22201 + \u2220x = 180\u00b0
\n\u22201 + 50\u00b0 = 180\u00b0
\n\u22201 = 180\u00b0 – 50\u00b0
\n\u22201 = 130\u00b0
\nNow,
\n\u2220x + \u22203 = 180\u00b0
\n\u22203 + 50\u00b0 = 180\u00b0
\n\u22203 = 180\u00b0 – 50\u00b0
\n\u22203 = 130\u00b0
\nHence, from the above,
\nWe can conclude that
\n\u22201 = \u22203<\/p>\n

Assessment Practice<\/strong>
\nQuestion 16.
\nUsing the diagram at the right, Martin incorrectly writes m\u2220b = 125\u00b0 What mistake did Martin likely make? Find the correct measure of \u2220b.
\n\"Envision
\nAnswer:
\nIt is given that
\nMartin incorrectly writes
\nm\u2220b = 125\u00b0
\nNow,
\nThe given figure is:
\n\"Envision
\nNow,
\nFrom the give figure,
\nWe can observe that \u2220b and 55\u00b0 are complementary angles
\nNow,
\nWe know that,
\nThe sum of complementary angles is always 90\u00b0
\nSo,
\nBoth the angles in the complementary angles must be acute angles i.e., the angle must be less than 90\u00b0
\nNow,
\n\u2220b + 55\u00b0 = 90\u00b0
\n\u2220b = 90\u00b0 – 55\u00b0
\n\u2220b = 35\u00b0
\nHence, from the above,
\nWe can conclude that
\nThe mistake made by Martin is: Consideration of the angles as supplementary angles even though they are complementary angles
\nThe correct measure of \u2220b is: 35\u00b0<\/p>\n

Question 17.
\nWhat is the measure, in degrees, of angle x?
\n\"Envision
\nAnswer:
\nThe given figure is:
\n\"Envision
\nNow,
\nFrom the given figure,
\nWe can observe that
\nx\u00b0 and 107\u00b0 are adjacent angles
\nNow,
\nWe know that,
\nThe sum of the adjacent angles is always 180\u00b0
\nSo,
\nx\u00b0 + 107\u00b0 = 180\u00b0
\nx\u00b0 = 180\u00b0 – 107\u00b0
\nx\u00b0= 73\u00b0
\nHence, from the above,
\nWe can conclude that the value of x is: 73\u00b0<\/p>\n

Question 18.
\nWhat is the measure, in degrees, of the highlighted angle?
\n\"Envision
\nAnswer:
\nThe given figure is:
\n\"Envision
\nNow,
\nFrom the given figure,
\nWe can observe that
\nx\u00b0 and 2x\u00b0 are the supplementary angles
\nNow,
\nWe know that,
\nThe sum of the supplementary angles is always 180\u00b0
\nSo,
\nx\u00b0 + 2x\u00b0 = 180\u00b0
\n3x\u00b0 = 180\u00b0
\nx\u00b0 = \\(\\frac{180\u00b0}{3}\\)
\nx\u00b0= 60\u00b0
\nSo,
\n2x\u00b0 = 2 (60\u00b0) = 120\u00b0
\nHence, from the above,
\nWe can conclude that the value of the highlighted angle is: 120\u00b0<\/p>\n

Lesson 8.5 Solve Problems Involving Circumference of a Circle<\/h3>\n

Explore It!<\/strong><\/p>\n

The distance around a circle and the distance across a circle are related.
\n\"Envision
\nI can… solve problems involving radius, diameter, and circumference of circles.<\/p>\n

A. Use string to measure the distance across each circle. How many of these lengths does it take to go completely around the circle?
\nAnswer:
\nWe know that,
\nThe distance around the circle is: Circumference
\nThe distance across the circle is: Diameter
\nNow,
\nThe complete length to go completely around the circle is given as:
\nCircumference = \u03c0d
\nHence, from the above,
\nWe can conclude that
\nThe total length does it take to go completely around the circle = \u03c0d
\nWhere
\nd is the diameter<\/p>\n

B. Use the string and a ruler to measure the distance across the circle and the distance around the circle. Complete the table. Round each measurement to the nearest quarter inch.
\n\"Envision
\nAnswer:<\/p>\n

C. What do you notice about the ratio of the distance around the circle to the distance across the circle for each circle?
\nAnswer:
\nFrom the given table,
\nWe can observe that the ratio of the circumference and the diameter is constant for all the circles
\nHence, from the above,
\nWe can conclude that the ratio of the distance around the circle to the distance across the circle for each circle is constant<\/p>\n

Focus on math practices<\/strong>
\nLook for Relationships How can you estimate the distance around any circle when given the distance across the circle?
\nAnswer:
\nWe know that,
\nThe distance around the circle is: Circumference
\nThe distance across the circle is: Diameter
\nNow,
\nWe know that,
\nThe relationship between the distance around the circle and the distance across the circle is:
\nDistance around the circle = \u03c0 \u00d7 Distance across the circle
\nHence, from the above,
\nWe can conclude that
\nCircumference (C) = \u03c0d
\nWhere,
\nd is the diameter<\/p>\n

Essential Question
\n<\/strong> How is the circumference of a circle related to the length of its diameter?
\nAnswer:
\nThe relationship between the circumference of a circle and the diameter of the circle is:
\nCircumference (C) = \u03c0d
\nWhere,
\nd is the diameter<\/p>\n

Try It!<\/strong>
\nWhat is the circumference of the rim of a basketball hoop with a radius of 9 inches?
\n\"Envision
\nFirst, multiply the radius by __________ to get the diameter, __________ inches. Then, multiply the diameter by 3.14 (an approximation for \u03c0) to get a circumference of about __________ inches.
\nAnswer:
\nThe given figure is:
\n\"Envision
\nNow,
\nWe know that,
\nThe diameter of the rim (d) = 2 \u00d7 Radius of the rim (r)
\nSo,
\nd = 2 \u00d7 9
\nd = 18 in.
\nNow,
\nWe know that,
\nCircumference (C) = \u03c0d
\nSo,
\nC = 3.14 \u00d7 18
\n= 56.52 in.
\nHence, from the above,
\nWe can conclude that the circumference of the rim of a basketball hoop is about 56.52 in.<\/p>\n

Convince Me! If the diameter is doubled, what happens to the circumference? Explain.
\nAnswer:
\nWe know that,
\nCircumference (C) = \u03c0d
\nSo,
\nWhen the diameter is doubled,
\nCircumference (C) = \u03c0 \u00d7 2d
\nHence, from the above,
\nWe can conclude that
\nIf the diameter is doubled, then the circumference will also be doubled<\/p>\n

Try It!<\/strong>
\nThe circle has a circumference of 9.42 units. What is the area of the square? Use 3.14 for \u03c0. Explain how you found the answer.
\n\"Envision
\nAnswer:
\nIt is given that the circle has a circumference of 9.42 units
\nNow,
\nThe given figure is:
\n\"Envision
\nNow,
\nFrom the given figure,
\nWe can observe that
\nThe radius of the circle = The side of the square
\nNow,
\nWe know that,
\nThe circumference of a circle (C) = 2\u03c0r
\nSo,
\n2\u03c0r = 9.42
\nr = \\(\\frac{9.42}{2\u03c0}\\)
\nr = 1.5 units
\nSo,
\nThe side of the square is: 1.5 units
\nNow,
\nWe know that,
\nThe area of the square = Side\u00b2
\nSo,
\nThe area of the square = (1.5)\u00b2
\n= 2.25 units\u00b2
\nHence, from the above,
\nWe can conclude that the area of the square is: 2.25 units\u00b2<\/p>\n

KEY CONCEPT<\/strong>
\nThe parts of a circle and their relationships are summarized in the diagram below.
\n\"Envision<\/p>\n

Do You Understand?<\/strong><\/p>\n

Question 1.
\nEssential Question How is the circumference of a circle related to the length of its diameter?
\nAnswer:
\nThe relationship between the circumference of a circle and the diameter of the circle is:
\nCircumference (C) = \u03c0d
\nWhere,
\nd is the diameter<\/p>\n

Question 2.
\nConstruct Arguments Are there any circles for which the relationship between the diameter and circumference cannot be represented by \u03c0? Explain.
\nAnswer:
\nWe know that,
\nThe circumference of a circle (C) = \u03c0d
\nWhere,
\nd is the diameter
\nNow,
\nWe will find the circumference of any circle (Semi circle, Quarter circle, etc.) from the above formula only with some modifications
\nHence, from the above,
\nWe can conclude that there are not any circles for which the relationship between the diameter and circumference cannot be represented by \u03c0<\/p>\n

Question 3.
\nBe Precise Can you find the exact circumference of a circle when you multiply the diameter by \\(\\frac{22}{7}\\)? Explain.
\nAnswer:
\nNo, we can not find the exact circumference of a circle when you multiply the diameter by \\(\\frac{22}{7}\\) because the diameter will not always be the multiple of 7 so that the circumference will be an integer<\/p>\n

Do You Know How?<\/strong><\/p>\n

Question 4.
\nWhat is the circumference of a circle with a radius of 5 inches?
\nAnswer:
\nThe given data is:
\nThe radius of a circle (r) = 5 inches
\nNow,
\nWe know that,
\nThe circumference of a circle (C) = 2\u03c0r
\nSo,
\nC = 2 \u00d7 3.14 \u00d7 5
\n= 31.4 inches
\nHence, from the above,
\nWe can conclude that the circumference of the circle with the given radius is: 31.4 inches<\/p>\n

Question 5.
\nWhat is the diameter of a circle with a circumference of 10.99 feet?
\nAnswer:
\nThe given data is:
\nThe circumference of a circle (C) = 10.99 feet
\nNow,
\nWe know that,
\nThe circumference of a circle (C) = \u03c0d
\nSo,
\n10.99 = 3.14 \u00d7 d
\nd = \\(\\frac{10.99}{3.14}\\)
\nd = 3.5 feet
\nHence, from the above,
\nWe can conclude that the diameter of the circle with the given circumference is: 3.5 feet<\/p>\n

Question 6.
\nHow many full revolutions does a car tire with a diameter of 25 inches make when the car travels one mile?
\n\"Envision
\nAnswer:
\nIt is given that
\nA car tire has a diameter of 25 inches
\nNow,,
\nWe have to find the circumference of the tire and the speed of the car to find the number of revolutions
\nNow,
\nWe know that,
\nThe circumference of a circle (C) = \u03c0d
\nSo,
\nC = 3.14 \u00d7 25
\n= 78.5 inches
\nSo,
\nThe circumference of the given car tire is: 78.5 inches
\nNow,
\nWe know that,
\n1 mile = 5,280 feet
\n1 feet = 12 inches
\nSo,
\n1 mile = 5,280 \u00d7 12
\n= 63,360 inches
\nNow,
\nWe know that,
\nThe number of revolutions the car tire can make = \\(\\frac{Speed of a car}{The circumference of a car tire}\\)
\n= \\(\\frac{63,360}{78.5}\\)
\n\u2245 807 revolutions
\nHence, from the above,
\nWe can conclude that
\nThe number of revolutions the given car tire can make with the given diameter is about 807 revolutions<\/p>\n

Practice & Problem Solving<\/strong><\/p>\n

Question 7.
\nFind the circumference of the circle. Use \u03c0 as part of the answer.
\n\"Envision
\nAnswer:
\nThe given figure is:
\n\"Envision
\nNow,
\nFrom the given figure,
\nWe can observe that
\nThe diameter of a circle (d) = 7 cm
\nNow,
\nWe know that,
\nThe circumference of a circle (C) = \u03c0d
\nSo,
\nC = 7\u03c0 cm
\nHence, from the above,
\nWe can conclude that the circumference of the circle with the given diameter in terms of \u03c0 is: 7\u03c0 cm<\/p>\n

Question 8.
\nFind the circumference of the circle. Use 3.14 for \u03c0. Round to the nearest hundredth.
\n\"Envision
\nAnswer:
\nThe given figure is:
\n\"Envision
\nNow,
\nFrom the given figure,
\nWe can observe that
\nThe diameter of a circle (d) = 20 ft
\nNow,
\nWe know that,
\nThe circumference of a circle (C) = \u03c0d
\nSo,
\nC = 3.14 \u00d7 20
\n= 62.80 ft
\nHence, from the above,
\nWe can conclude that the circumference of the circle with the given diameter is: 62.80 ft<\/p>\n

Question 9.
\nFind the circumference of the circle. Use \u03c0 as part of the answer.
\n\"Envision
\nAnswer:
\nThe given figure is:
\n\"Envision
\nNow,
\nFrom the given figure,
\nWe can observe that
\nThe radius of a circle (r) = 12 mi
\nNow,
\nWe know that,
\nThe circumference of a circle (C) = 2\u03c0r
\nSo,
\nC = 2\u03c0 \u00d7 12
\n= 24\u03c0 mi
\nHence, from the above,
\nWe can conclude that
\nThe circumference of the circle with the given radius in terms of \u03c0 is: 24\u03c0 mi<\/p>\n

Question 10.
\nFind the circumference of the circle. Use 3.14 for \u03c0. Round to the nearest hundredth.
\n\"Envision
\nAnswer:
\nThe given figure is:
\n\"Envision
\nNow,
\nFrom the given figure,
\nWe can observe that
\nThe radius of a circle (r) = 9.5 cm
\nNow,
\nWe know that,
\nThe circumference of a circle (C) = 2\u03c0r
\nSo,
\nC = 2 \u00d7 3.14 \u00d7 9.5
\n= 59.66 cm
\nHence, from the above,
\nWe can conclude that the circumference of the circle with the given radius is: 59.66 cm<\/p>\n

Question 11.
\nFind the diameter of a circle with a circumference of 27 centimeters. Use 3.14 for \u03c0. Round to the nearest tenth.
\nAnswer:
\nThe given data is:
\nThe circumference of a circle (C) = 27 cm
\nNow,
\nWe know that,
\nThe circumference of a circle (C) = \u03c0d
\nSo,
\n27 = 3.14 \u00d7 d
\nd = \\(\\frac{27}{3.14}\\)
\nd = 8.59
\nd \u2245 9 cm
\nHence, from the above,
\nWe can conclude that the diameter of the circle with the given circumference is about 9 cm<\/p>\n

Question 12.
\nThe distance around a meteor crater is 9,687 feet. Find the diameter of the crater. Use \\(\\frac{22}{7}\\) for \u03c0. Round to the nearest tenth.
\n\"Envision
\nAnswer:
\nIt is given that
\nThe distance around a meteor crater is 9,687 feet
\nNow,
\nThe given figure is:
\n\"Envision
\nNow,
\nFrom the above,
\nWe can observe that the crater is in the form of a circle
\nNow,
\nWe know that,
\nThe distance around the circle is: Circumference
\nNow,
\nWe know that,
\nThe circumference of a circle (C) = \u03c0d
\nSo,
\n9,687 = \\(\\frac{22}{7}\\) \u00d7 d
\nd = \\(\\frac{9,687 \u00d7 7}{22}\\)
\nd = 3,082.2 feet
\nHence, from the above,
\nWe can conclude that the diameter of the crater for the given circumference of a crater is: 3,082.2 feet<\/p>\n

Question 13.
\nMake Sense and Persevere The circumference of the inner circle is 44 feet. The distance between the inner circle and the outer circle is 3 feet. By how many feet is the circumference of the outer circle greater than the circumference of the inner circle? Use \\(\\frac{22}{7}\\) for \u03c0. Round to the nearest hundredth of a foot.
\n\"Envision
\nAnswer:
\nIt is given that
\nThe circumference of the inner circle is 44 feet. The distance between the inner circle and the outer circle is 3 feet
\nNow,
\nWe know that,
\nThe circumference of inner circle (c) = 2\u03c0r
\nSo,
\nFor inner circle:
\n44 = 2 \u00d7 \\(\\frac{22}{7}\\) \u00d7 r
\nr = \\(\\frac{44 \u00d7 7}{22 \u00d7 2}\\)
\nr = 7 feet
\nFor outer circle:
\nThe circumference of the outer circle (C) = 2\u03c0(R – r)
\nWhere,
\nR is the radius of the outer circle
\nr is the radius of the inner circle
\nSo,
\nC = 2 \u00d7 \\(\\frac{22}{7}\\) \u00d7 (7 + 3)
\nC = 62.85 feet
\nSo,
\nC – c = 62.85 – 44
\n= 18.85 feet
\nHence, from the above,
\nWe can conclude that the circumference of the outer circle is 18.85 feet greater than the circumference of the inner circle<\/p>\n

Question 14.
\nGeneralize What is the ratio of the radius to the circumference of any circle, using 3.14 for \u03c0?
\nAnswer:
\nWe know that,
\nThe circumference of a circle (C) = 2\u03c0r
\nSo,
\n\\(\\frac{C}{r}\\) = 2\u03c0
\n\\(\\frac{C}{r}\\) = 6.28
\nHence, from the above,
\nWe can conclude that the ratio of the radius to the circumference of any circle is: 6.28<\/p>\n

Question 15.
\nWhat is the radius of a circle with a circumference of 26.69 centimeters?
\nAnswer:
\nThe given data is:
\nThe circumference of a circle (C) = 26.69 cm
\nNow,
\nWe know that,
\nThe circumference of a circle (C) = 2\u03c0r
\nSo,
\n26.69 = 2 \u00d7 3.14 \u00d7 r
\nr = \\(\\frac{26.69}{6.28}\\)
\nr = 4.25 cm
\nHence, from the above,
\nWe can conclude that the radius of the circle with the given circumference is: 4.25 cm<\/p>\n

Question 16.
\nHigher Order Thinking A unicycle wheel makes five rotations. The unicycle travels 37.94 feet. Find the diameter of the wheel in inches. Use 3.14 for \u03c0. Round to the nearest tenth of an inch.
\n\"Envision
\nAnswer:
\nIt is given that
\nA unicycle wheel makes five rotations. The unicycle travels 37.94 feet
\nNow,
\nWe know that,
\nThe number of revolutions (or) rotations = \\(\\frac{The speed of the unicycle}{The circumference of the unicycle wheel}\\)
\nSo,
\nThe circumference of the unicycle wheel = \\(\\frac{The speed of the unicycle}{The number of rotations}\\)
\n3.14 \u00d7 d = \\(\\frac{37.94}{5}\\)
\n3.14 \u00d7 d = 7.58
\nd = \\(\\frac{7.58}{3.14}\\)
\nd = 2.41 feet
\nd = 2.41 \u00d7 12
\nd = 29 inches
\nHence, from the above,
\nWe can conclude that the diameter of the wheel in inches is: 29 inches<\/p>\n

Assessment Practice<\/strong>
\nQuestion 17.
\nCamille drew the figure shown at the right. Which of the following is the best estimate of the perimeter of the figure?
\n\"Envision
\nA. 36 feet
\nB. 81 feet
\nC. 45 feet
\nD. 50 feet
\nAnswer:<\/p>\n

Question 18.
\nA cabin on a Ferris wheel has traveled one-fourth of the circumference of the wheel, a distance of 117.75 feet. What is the radius, in feet, of the Ferris wheel? Use 3.14 for \u03c0.
\nAnswer:
\nIt is given that
\nA cabin on a Ferris wheel has traveled one-fourth of the circumference of the wheel, a distance of 117.75 feet
\nNow,
\nThe circumference of the wheel (C) = 4 \u00d7 117.75
\n= 471 feet
\nNow,
\nWe know that,
\nThe circumference of a circle (C) = 2\u03c0r
\nSo,
\n471 = 2 \u00d7 3.14 \u00d7 r
\nr = \\(\\frac{471}{6.28}\\)
\nr = 75 feet
\nHence, from the above,
\nWe can conclude that the radius of the Ferris wheel is: 75 feet<\/p>\n

Question 19.
\nThe diagram shows a track composed of a rectangle with a semicircle on each end. The area of the rectangle is 7,200 square meters. What is the perimeter, in meters, of the track? Use 3.14 for \u03c0.
\n\"Envision
\nAnswer:
\nIt is given that
\nThe diagram shows a track composed of a rectangle with a semicircle on each end. The area of the rectangle is 7,200 square meters.
\nNow,
\nThe given figure is:
\n\"Envision
\nNow,
\nWe know that,
\nThe area of the rectangle (A) = Length \u00d7 Width
\nSo,
\n7,200 = 120 \u00d7 Width
\nWidth = \\(\\frac{7,200}{120}\\)
\nWidth = 60 m
\nSo,
\nThe radius of the semicircle (r) = \\(\\frac{Width of the rectangle}{2}\\)
\nr = \\(\\frac{60}{2}\\)
\nr = 30 m
\nNow,
\nWe know that,
\nThe perimeter of a rectangle = 2 (Length + Width)
\nThe perimeter of a semicircle = r (\u03c0 + 2)
\nNow,
\nThe perimeter of a rectangle = 2 (120 + 60)
\n= 2 (180)
\n= 360 m
\nThe perimeter of a semicircle = 30 (3.14 + 2)
\n= 154.2 m
\nSo,
\nThe perimeter of the track = The perimeter of a rectangle + 2 (The perimeter of a semicircle)
\n= 360 + 2 (154.2)
\n= 668.4 m
\nHence, from the above,
\nWe can conclude that the perimeter of the given track is: 668.4 m<\/p>\n

Topic 8 MID-TOPIC CHECKPOINT<\/h3>\n

Question 1.
\nVocabulary How are adjacent angles and vertical angles alike? How are they different? Lesson 8-4
\nAnswer:
\nWe know that,
\nThe adjacent angles and the vertical angles have the same vertex
\nWe know that,
\nThe sum of the adjacent angles is 180\u00b0 (or) 90\u00b0<\/p>\n

Question 2.
\nOn a map, 1 inch equals 150 miles. The border between the two states is 5.5 inches long on the map. What is the actual length of the border? Lesson 8-1
\nAnswer:
\nIt is given that
\nOn a map, 1 inch equals 150 miles. The border between the two states is 5.5 inches long on the map.
\nSo,
\nThe actual length of the border – (Scale factor) \u00d7 (The border between the two states on the amp)
\n= 150 \u00d7 5.5
\n= 825 miles
\nHence, from the above,
\nWe can conclude that the actual length of the border is: 825 miles<\/p>\n

In 3 and 4, use the figure to the right.
\n\"Envision
\nQuestion 3.
\nWhat is the measure of \u2220BZD? Lesson 8-4
\nA. 58\u00b0
\nB. 148\u00b0
\nC. 32\u00b0
\nD. 90\u00b0
\nAnswer:
\nThe given figure is:
\n\"Envision
\nNow,
\nFrom the given figure,
\nWe can observe that
\n\u2220BZD = \u2220BZC + \u2220CZD
\nSo,
\n\u2220BZD = 58\u00b0 + 90\u00b0
\n\u2220BZD = 148\u00b0
\nHence, from the above,
\nWe can conclude that
\nThe angle measure of\u00a0\u2220BZD is:
\n\"\"<\/p>\n

Question 4.
\nFind the value of x. Lesson 8-4
\nAnswer:
\nThe given figure is:
\n\"Envision
\nNow,
\nFrom the given figure,
\nWe can observe that
\n(2x)\u00b0 + 58\u00b0 = 180\u00b0
\n(2x)\u00b0 = 180\u00b0 – 58\u00b0
\n(2x)\u00b0 = 122\u00b0
\nx = \\(\\frac{122\u00b0}{2}\\)
\nx = 61\u00b0
\nHence,f rom the above,
\nWe can conclude that the value of x is: 61\u00b0<\/p>\n

Question 5.
\nPierce draws a circle with a radius of 3 centimeters. Gianna draws a circle with a radius that is twice as long as the radius of Pierce’s circle. How will the circumference of Gianna’s circle compare with the circumference of Pierce’s circle? Lesson 8-5
\nThe circumference of Gianna’s circle is ___________ times the circumference of Pierce’s circle.
\nAnswer:
\nIt is given that
\nPierce draws a circle with a radius of 3 centimeters. Gianna draws a circle with a radius that is twice as long as the radius of Pierce’s circle
\nSo,
\nThe radius of Pierce’s circle is: 3 cm
\nSo,
\nThe radius of Gianna’s circle = 2 \u00d7 (Radius of Pierce’s circle)
\n= 2 \u00d7 3
\n= 6 cm
\nNow,
\nThe circumference of Pierce’s circle (C) = 2\u03c0r
\n= 2 \u00d7 \u03c0 \u00d7 3
\n= 6\u03c0 cm
\nNow,
\nThe circumference of Gianna’s circle (C) = 2\u03c0r
\n= 2 \u00d7 \u03c0 \u00d7 6
\n= 12\u03c0 cm
\nNow,
\nLet the number of times Gianna’s circumference is a number of times as Pierce’s circumference be x
\nSo,
\nx \u00d7 6\u03c0 = 12\u03c0
\nx = \\(\\frac{12\u03c0}{6\u03c0}\\)
\nx = 2
\nHence, from the above,
\nWe can conclude that
\nThe circumference of Gianna’s circle is 2 times the circumference of Pierce’s circle.<\/p>\n

Question 6.
\nDraw a triangle with one side length of 5 units and another side length of 7 units. What additional piece of information will guarantee that only one triangle can be drawn? Lessons 8-2 and 8-3
\nAnswer:
\nThe representation of the triangle with the given side lengths is:
\n\"\"
\nNow,
\nThe additional information that will guarantee that only one triangle can be drawn is:
\nThe angle between the two given side lengths i.e., an included angle<\/p>\n

Topic 8 MID-TOPIC PERFORMANCE TASK<\/h3>\n

Mrs. Thomas has two rolls of garden edging that are each 96 inches long. She wants to make two new flower beds in her backyard. Each flower bed will be bordered by one roll of edging. One flower bed will be in the shape of a quadrilateral. The other will be in the shape of a triangle.<\/p>\n

PART A
\nMrs. Thomas decides to make a scale drawing of each flower bed using a scale of 1 centimeter = 5 inches. What will be the total length of each roll of edging in her scale drawings?
\nAnswer:
\nIt is given that
\nMrs. Thomas has two rolls of garden edging that are each 96 inches long. She wants to make two new flower beds in her backyard. Each flower bed will be bordered by one roll of edging. One flower bed will be in the shape of a quadrilateral. The other will be in the shape of a triangle.
\nMrs. Thomas decides to make a scale drawing of each flower bed using a scale of 1 centimeter = 5 inches.
\nNow,
\nFrom the given information,
\nScale factor = \\(\\frac{5}{1}\\)
\nSo,
\nThe total length of each roll of edging in her scale drawings = \\(\\frac{The total length of each roll of edging}{Scale factor}\\)
\n= \\(\\frac{96 \u00d7 2}{5}\\)
\n= 19.2 \u00d7 2
\n= 38.4 inches
\nHence, from the above,
\nWe can conclude that
\nThe total length of each roll of edging in her scale drawings is: 38.4 inches<\/p>\n

PART B
\nMrs. Thomas wants the quadrilateral flower bed to have at least two 90\u00b0 angles. Draw a possible plan for this flower bed using the scale from Part A. Make sure to use a complete roll of edging in the border. Label your drawing with all the angle measures and with the scaled length of each side. Name the shape of the flower bed you drew. What will be its actual dimensions?
\nAnswer:
\nIt is given that
\nMrs. Thomas wants the quadrilateral flower bed to have at least two 90\u00b0 angles
\nNow,
\nFrom part (a),
\nThe scale is: 1 cm = 5 inches
\nSo,
\nThe representation of the quadrilateral flower bed with the scaled measurements is:
\n\"\"
\nNow,
\nThe actual dimensions of the given quadrilateral flower bed are:
\nThe length of the quadrilateral flower bed = (Scaled length) \u00d7 (Scale factor)
\n= 4 \u00d7 5
\n= 20 inches
\nSince the quadrilateral bed is a square bed, all the side lengths are equal
\nHence, from the above,
\nWe can conclude that
\nThe shape of the flower bed is: Square
\nThe actual dimensions of the flower bed are: 20 inches, 20 inches, 20 inches, 20 inches<\/p>\n

PART C
\nMrs. Thomas began to make a drawing for the triangular flower bed. In her drawing, the length of one side of the triangle is 4.8 centimeters, the length of the second side is 6.4 centimeters, and the included angle is a right angle. Use these measures and the scale from Part A to make a completed scale drawing. Label your drawing with all the angle measures to the nearest whole degree and with the scale length of each side. What will be the actual dimensions of this flower bed?
\nAnswer:
\nIt is given that
\nMrs. Thomas began to make a drawing for the triangular flower bed. In her drawing, the length of one side of the triangle is 4.8 centimeters, the length of the second side is 6.4 centimeters, and the included angle is a right angle
\nNow,
\nFrom part (a),
\nThe scale is: 1 cm = 5 inches
\nNow,
\nThe representation of the triangular flower bed with the given specifications is:
\n\"\"
\nNow,
\nThe actual dimensions of the triangular flower garden are:
\nThe actual base length of the flower garden = (The scaled base length of the flower garden) \u00d7 (Scale factor)
\n= 6.4 \u00d7 5
\n= 32.0
\n= 32 inches
\nThe actual height of the flower garden = (The scaled height of the flower garden) \u00d7 (scale factor)
\n= 4.8 \u00d7 5
\n= 24.0
\n= 24 inches
\nThe actual hypotenuse length of the flower garden = (The scaled hypotenuse length of the flower garden) \u00d7 (Scale factor)
\n= 8 \u00d7 5
\n= 40 inches
\nHence, from the above,
\nWe can conclude that
\nThe actual dimensions of the flower garden are:
\nBase length: 32 inches
\nHeight: 24 inches
\nHypotenuse: 40 inches<\/p>\n

Lesson 8.6 Solve Problems Involving Area of a Circle<\/h3>\n

Explore It!<\/strong><\/p>\n

Latoya cut a circle into 8 equal sections and arranged the pieces to form a shape resembling a parallelogram.
\n\"Envision
\nI can… solve problems involving the area of a circle.<\/p>\n

A. How is the base length of the new shape related to the circumference of the circle?
\nAnswer:
\nIt is given that
\nLatoya cut a circle into 8 equal sections and arranged the pieces to form a shape resembling a parallelogram.
\nNow,
\nThe given figure is:
\n\"Envision
\nNow,
\nFrom the given figure,
\nWe can observe that
\nThe base of the new figure is equal to half of the value of the circumference of the circle
\nHence, from the above,
\nWe can conclude that
\nThe base of the new figure = \\(\\frac{1}{2}\\) \u00d7 Circumference of the circle<\/p>\n

B. How is the height of the new shape related to the radius of the circle?
\nAnswer:
\nThe given figure is:
\n\"Envision
\nNow,
\nFrom the given figure,
\nWe can observe that the height of the new figure is equal to the radius of the circle
\nHence, from the above,
\nWe can conclude that
\nThe height of the new figure = Radius of the circle<\/p>\n

C. Since this new shape was made from a circle, use the information from the diagram and the formula for the area of the parallelogram, A= bh, to discover the formula for the area of a circle.
\nAnswer:
\nIt is given that
\nThe area of the parallelogram (A) = bh
\nWhere,
\nb is the Base of the parallelogram
\nh is the height of the parallelogram
\nBut,
\nIt is given that
\nThe parallelogram is made from a circle
\nNow,
\nFrom part (a) and (b),
\nA = \\(\\frac{1}{2}\\) \u00d7 The circumference of a circle \u00d7 Radius of a circle
\nA = \\(\\frac{1}{2}\\) \u00d7 2\u03c0r \u00d7 r
\nA = \u03c0r \u00d7 r
\nA = \u03c0r\u00b2
\nHence, from the above,
\nWe can conclude that teh area of the circle that is derived fro the area of the parallelogram is: \u03c0r\u00b2<\/p>\n

Focus on math practices<\/strong>
\nLook for Relationships The formula A = bh can be used to find a good estimate for the area of the cut-out diagram. What would happen to this estimate if the circle was cut into 100 sections? 1,000 sections?
\nAnswer:
\nThe given figure is:
\n\"Envision
\nNow,
\nFrom the given figure,
\nWe can observe that
\nThe 8 parts of the circle made 4 parallelograms
\nSo,
\nThe relationship between the circles and parallelograms is:
\nThe number of parallelograms = \\(\\frac{The number of parts of the circle}{2}\\)
\nSo,
\nFor 100 pieces of the circle,
\nThe number of parallelograms = \\(\\frac{100}{2}\\)
\n= 50
\nFor 1,000 pieces of the circle,
\nThe number of parallelograms = \\(\\frac{1,000}{2}\\)
\n= 500
\nHence,
\nFor 100 pieces of the circle,
\nThe area of the parallelogram (A) = Base \u00d7 Height [Since base = \\(\\frac{1}{2}\\) \u00d7 Circumference]
\n= 50r units\u00b2
\nFor 1,000 pieces of the circle,
\nThe area of the parallelogram (A) = Base \u00d7 Height
\n= 500r units\u00b2<\/p>\n

Essential Question<\/strong>
\nHow can the area formula for a circle be used to solve problems?
\nAnswer:
\nArea of a circle can be calculated by using the formulas:
\na. Area = \u03c0 \u00d7 r2<\/sup>
\nwhere,
\n‘r’ is the radius.
\nb. Area = (\u03c0\/4) \u00d7 d2<\/sup>
\nwhere,
\n‘d’ is the diameter.
\nc. Area = C2<\/sup>\/4\u03c0
\nwhere,
\n‘C’ is the circumference<\/p>\n

Try It!<\/strong>
\nAt a school play, there is a spotlight above the center of the floor that covers a lighted area with a radius of 7 feet. What is the area covered by the spotlight?
\n\"Envision
\nThe area covered by the spotlight is about _________ square feet.
\nAnswer:
\nIt is given that
\nAt a school play, there is a spotlight above the center of the floor that covers a lighted area with a radius of 7 feet
\nNow,
\nWe know that,
\nThe area of a circle (A) = \u03c0r\u00b2
\nSo,
\nThe area covered by the spotlight (A) = \\(\\frac{22}{7}\\) \u00d7 7\u00b2
\nA = 154 feet\u00b2
\nHence, from the above,
\nWe can conclude that the area covered by the spotlight is: 154 square feet<\/p>\n

Convince Me! If the diameter of a circle is given, how would you find the area?
\nAnswer:
\nIf the diameter of a circle is given, then
\nThe area of a circle = (\u03c0\/4) \u00d7 d2<\/sup>
\nwhere,
\n‘d’ is the diameter.<\/p>\n

Try It!<\/strong>
\na. How far away can a person live from a radio station and hear its broadcast if the signal covers a circular area of 40,000 square miles? Write your answer as a whole number.
\nAnswer:
\nIt is given that
\nA person live from a radio station and hear its broadcast if the signal covers a circular area of 40,000 square miles
\nNow,
\nWe have to find the diameter of a circle to find how far a person lives from a radio station
\nNow,
\nWe know that,
\nThe area of a circle (A) = \u03c0r\u00b2
\nSo,
\n40,000 = 3.14 \u00d7 r\u00b2
\nr\u00b2 = \\(\\frac{40,000}{3.14}\\)
\nr\u00b2 = 12,738.85
\nr = 112.8
\nr \u2248 113 miles
\nSo,
\nThe diameter of a circle = 2 \u00d7 The radius of a circle
\n= 2 \u00d7 113
\n= 226 miles
\nHence, from the above,
\nWe can conclude that the distance a person can live from a radio station and hear its broadcast is: 226 miles<\/p>\n

b. What circular area is covered by the signal if the circumference is 754 miles?
\nAnswer:
\nIt is given that
\nThe circumference of the circular signal is 754 miles
\nNow,
\nWe know that,
\nThe circumference of a circle (C) = 2\u03c0r
\nSo,
\n754 = 2 \u00d7 3.14 \u00d7 r
\nr = \\(\\frac{754}{6.28}\\)
\nr = 120 miles
\nNow,
\nThe area of the circular signal (A) = \u03c0r\u00b2
\nSo,
\nA = 3.14 \u00d7 (120)\u00b2
\nA = 45,216 miles\u00b2
\nHence, from the above,
\nWe can conclude that teh circular area of the given signal is: 45,216 miles\u00b2<\/p>\n

KEY CONCEPT<\/strong><\/p>\n

You can find the area, A, of a circle using the formula A = \u03c0r2<\/sup>, where r is the radius.
\n\"Envision<\/p>\n

Do You Understand?<\/strong><\/p>\n

Question 1.
\nEssential Question How can the area formula for a circle be used to solve problems?
\nAnswer:
\nArea of a circle can be calculated by using the formulas:
\na. Area = \u03c0 \u00d7 r2<\/sup>
\nwhere,
\n‘r’ is the radius.
\nb. Area = (\u03c0\/4) \u00d7 d2<\/sup>
\nwhere,
\n‘d’ is the diameter.
\nc. Area = C2<\/sup>\/4\u03c0
\nwhere,
\n‘C’ is the circumference<\/p>\n

Question 2.
\nBe Precise<\/strong>
\nIs an area calculation exact when you use 3.14 or \\(\\frac{22}{7}\\) as a value for \u03c0? Explain.
\nAnswer:
\nThe constant (pi) is not equal to either 22\/7 or 3.14; these are only approximations to . The actual value of is the\u00a0ratio of the circumference of a circle to its diameter, and is a transcendental number with infinitely many non-repeating decimal places<\/p>\n

Question 3.
\nUse Structure if you know the diameter of a circle, how can you find the area?
\nAnswer:
\nIf you know the diameter of a circle, then
\nThe area of a circle = (\u03c0\/4) \u00d7 d2<\/sup>
\nwhere,
\n‘d’ is the diameter.<\/p>\n

Do You Know How?<\/strong><\/p>\n

For 4-7, use 3.14 for \u03c0.
\nQuestion 4.
\nWhat is the area of a circle with a radius of 8 inches?
\nAnswer:
\nThe given data is:
\nThe radius of a circle is 8 inches
\nNow,
\nWe know that,
\nThe area of a circle (A) = \u03c0r\u00b2
\nSo,
\nA = 3.14 \u00d7 8\u00b2
\n= 200.96 inches\u00b2
\nHence, from the above,
\nWe can conclude that the area of the circle with the given radius is: 200.96 inches\u00b2<\/p>\n

Question 5.
\nWhat is the radius of a circle with an area of 28.26 square feet?
\nAnswer:
\nThe given data is:
\nThe area of a circle is: 28.26 square feet
\nNow,
\nWe know that,
\nThe area of a circl (A) = \u03c0r\u00b2
\nSo,
\n28.26 = 3.14 \u00d7 r\u00b2
\nr\u00b2 = \\(\\frac{28.26}{3.14}\\)
\nr\u00b2 = 9
\nr = 3 feet
\nHence, from the above,
\nWe can conclude that the radius of the circle with the given area is: 3 feet<\/p>\n

Question 6.
\nWhat is the area of a circle with a circumference of 25.12 meters?
\nAnswer:
\nThe given data is:
\nThe circumference of a circle is: 25.12 meters
\nNow,
\nWe know that,
\nThe circumference of a circle (C) = 2\u03c0r
\nSo,
\n25.12 = 2 \u00d7 3.14 \u00d7 r
\nr = \\(\\frac{25.12}{6.28}\\)
\nr = 4 meters
\nNow,
\nWe know that,
\nThe area of a circle (A) = \u03c0r\u00b2
\nSo,
\nA = 3.14 \u00d7 4\u00b2
\n= 50.24 sq. meters
\nHence, from the above,
\nWe can conclude that the area of the circle with the given circmference is: 50.24 sq.meters<\/p>\n

Question 7.
\nThe diameter of a pizza is 12 inches. What is its area?
\n\"Envision
\nAnswer:
\nIt is given that
\nThe diameter of a pizza is 12 inches.
\nNow,
\nThe given figure is:
\n\"Envision
\nNow,
\nWe know that,
\nRadius of a circle (r) = \\(\\frac{Diameter of a circle}{2}\\)
\nr = \\(\\frac{12}{2}\\)
\nr = 6 in.
\nNow,
\nWe know that,
\nThe area of a circle (A) = \u03c0r\u00b2
\nSo,
\nA = 3.14 \u00d7 6\u00b2
\n= 113.04 in.\u00b2
\nHence, from the above,
\nWe can conclude that the area of the circle with the given diameter is: 113.04 in.\u00b2<\/p>\n

Practice & Problem Solving<\/strong><\/p>\n

Question 8.
\nFind the area of the circle. Use 3.14 for \u03c0.
\n\"Envision
\nAnswer:
\nThe given figure is:
\n\"Envision
\nNow,
\nFrom the given figure,
\nWe can observe that
\nThe radius of a circle (r) = 9 ft
\nNow,
\nWe know that,
\nThe arae of a circle (A) = \u03c0r\u00b2
\nSo,
\nA = 3.14 \u00d7 9\u00b2
\n= 254.34 ft\u00b2
\nHence, from the above,
\nWe can conclude that teh arae of the circle with the given radius is: 254.34 feet\u00b2<\/p>\n

Question 9.
\nFind the area of the circle. Use 3.14 for \u03c0.
\n\"Envision
\nAnswer:
\nThe given figure is:
\n\"Envision
\nNow,
\nFrom the given figure,
\nWe can observe that
\nThe diameter of a circle is: 106 yd
\nNow,
\nWe know that,
\nThe radius of a circle (r) = \\(\\frac{The diameter of a circle}{2}\\)
\nr = \\(\\frac{106}{2}\\)
\nr = 53 yd
\nNow,
\nWe know that,
\nThe area of a circle (A) = \u03c0r\u00b2
\nSo,
\nA = 3.14 \u00d7 53\u00b2
\n= 8,820.26 yd\u00b2
\nHence, from the above,
\nWe can conclude that the area of the circle with the given diameter is: 8,820.26 yd\u00b2<\/p>\n

Question 10.
\nJaylon created this stained-glass window. The upper two corners are quarter circles, each with a radius of 4 inches. Find the area of the window. Use\u00a03.14 for \u03c0.
\n\"Envision
\nAnswer:
\nIt is given that
\nJaylon created this stained-glass window. The upper two corners are quarter circles, each with a radius of 4 inches.
\nNow,
\nThe given figure is:
\n\"Envision
\nNow,
\nFrom the given figure,
\nWe can observe that
\nThe stained glass window is made up of 2 quarter circles, 1 rectangle, and 1 square
\nNow,
\nWe know that,
\nThe area of a rectangle = Length \u00d7 Width
\nThe arae of a quarter circle = \\(\\frac{1}{4}\\)\u03c0r\u00b2
\nThe area of a square = Side \u00d7 Side
\nNow,
\nThe area of 2 quarter circles (A) = 2 \u00d7 \\(\\frac{1}{4}\\) \u00d7 3.14 \u00d7 4\u00b2
\n= 25.12 in.\u00b2
\nThe area of the recatngle (A) = 12 \u00d7 (26 – 4)
\n= 264 in.\u00b2
\nThe area of the square that lies between the two quareter circles (A) = (12 – 8) \u00d7 4
\n= 16 in.\u00b2
\nSo,
\nThe area of the stained glass window = 25.12 + 264 + 16
\n= 305.12 in.\u00b2
\nHence, from the above,
\nWe can conclude that the area of the stained glass window is: 305.12 in.\u00b2<\/p>\n

Question 11.
\nThe circumference of a circle is 50.24 meters. What is the area of the circle? Use 3.14 for \u03c0.
\nAnswer:
\nIt is given that
\nThe circumference of a circle is 50.24 meters
\nNow,
\nWe know that,
\nThe circumference of a circle (C) = 2\u03c0r
\nSo,
\n50.24 = 2 \u00d7 3.14 \u00d7 r
\nr = \\(\\frac{50.24}{6.28}\\)
\nr = 8 meters
\nNow,
\nWe know that,
\nThe area of a circle (A) = \u03c0r\u00b2
\nSo,
\nA = 3.14 \u00d7 8\u00b2
\n= 200.96 meters\u00b2
\nHence, from the above,
\nWe can conclude that the area of the circle for the given circumference is: 200.96 meters\u00b2<\/p>\n

Question 12.
\nHigher-Order Thinking<\/strong> A circular flower bed is 20 meters in diameter and has a circular sidewalk around it that is 3 meters wide. Find the area of the sidewalk in square meters. Use 3.14 for \u03c0. Round to the nearest whole number.
\nAnswer:
\nIt is given that
\nA circular flower bed is 20 meters in diameter and has a circular sidewalk around it that is 3 meters wide.
\nNow,
\nThe area of the sidewalk is given as:
\nA = The area of a flower bed – The area of a sidewalk including the flower bed
\nNow,
\nWe know that,
\nRadius (r) = \\(\\frac{Diameter}{2}\\)
\nr = \\(\\frac{20}{2}\\)
\nr = 10 meters
\nNow,
\nThe area of a flower bed (A) = \u03c0r\u00b2
\nSo,
\nA = 3.14 \u00d7 10\u00b2
\n= 314 meters\u00b2
\nNow,
\nThe area of a sidewalk including the flower bed (A) = 3.14 \u00d7 (10 + 3)\u00b2
\n= 530.66 meters\u00b2
\nSo,
\nThe area of the sidewalk = 530.66 – 314
\n= 216.66 meters\u00b2
\nHence, from the above,
\nWe can conclude that the area of the sidewalk is: 216.66 meters\u00b2<\/p>\n

Question 13.
\nA circular plate has a circumference of 16.3 inches. What is the area of this plate? Use 3.14 for \u03c0. Round to the nearest whole number.
\nAnswer:
\nIt is given that
\nA circular plate has a circumference of 16.3 inches
\nNow,
\nWe know that,
\nThe circumference of a circle (C) = 2\u03c0r
\nSo,
\n16.3 = 2 \u00d7 3.14 \u00d7 r
\nr = \\(\\frac{16.3}{6.28}\\)
\nr = 2.59 inches
\nNow,
\nWe know that,
\nThe area of the circular plate (A) = \u03c0r\u00b2
\nSo,
\nA = 3.14 \u00d7 (2.59)\u00b2
\n= 21.06
\n\u224821 inches\u00b2
\nHence, from the above,
\nWe can conclude that the area of the circular plate for the given circumference is: 21 inches\u00b2<\/p>\n

Question 14.
\nA water sprinkler sends water out in a circular pattern. How many feet away from the sprinkler can it spread water if the area formed by the watering pattern is 379.94 square feet?
\nAnswer:
\nIt is given that
\nA water sprinkler sends water out in a circular pattern and the area formed by the watering pattern is 379.94 square feet
\nNow,
\nWe have to find the diameter to find how many feet away from the sprinkler can spread the water
\nNow,
\nWe know that,
\nThe area of a circle (A) = \u03c0r\u00b2
\nSo,
\n379.94 = 3.14 \u00d7 r\u00b2
\nr\u00b2 = \\(\\frac{379.94}{3.14}\\)
\nr\u00b2 = 121
\nr = 11 feet
\nSo,
\nDiameter (d) = 2 \u00d7 Radius
\nd = 2 \u00d7 11
\nd = 22 feet
\nHence, from the above,
\nWe can conclude that the sprinkler can spread the water 22 feet away<\/p>\n

Question 15.
\nThe circumference of a circular rug is 24.8 meters. What is the area of the rug? Use 3.14 for \u03c0. Round your answer to the nearest tenth.
\n\"Envision
\nAnswer:
\nIt is given that
\nThe circumference of a circular rug is 24.8 meters
\nNow,
\nThe given figure is:
\n\"Envision
\nNow,
\nWe know that,
\nThe circumference of a circle (C) = 2\u03c0r
\nSo,
\n24.8 = 2 \u00d7 3.14 \u00d7 r
\nr = \\(\\frac{24.8}{6.28}\\)
\nr = 3.94 m
\nNow,
\nThe area of the circular rug (A) = \u03c0r\u00b2
\nSo,
\nA = 3.14 \u00d7 (3.94)\u00b2
\n= 48.7 m\u00b2
\nHence, from the above,
\nWe can conclude that the area of the circular rug for the given circumference to the nearest tenth is: 48.7 m\u00b2<\/p>\n

Question 16.
\nFrank wants to find the area enclosed by the figure at the right. The figure has semicircles on each side of a 40-meter-by-40-meter square. Find the area enclosed by the figure. Use 3.14 for \u03c0.
\n\"Envision
\nAnswer:
\nIt is given that
\nFrank wants to find the area enclosed by the figure at the right. The figure has semicircles on each side of a 40-meter-by-40-meter square.
\nNow,
\nThe given figure is:
\n\"Envision
\nNow,
\nThe area enclosed by the given figure = The area of the square + The area of the 4 semicircles
\nNow,
\nFrom the given figure,
\nWe can observe that
\nThe side of the square = 40 m
\nThe diameter of the semi circle = 40 m
\nNow,
\nWe know that,
\nThe area of a square (A) = Side\u00b2
\nSo,
\nA = 40\u00b2
\n= 1,600 m\u00b2
\nNow,
\nWe know that,
\nThe area of a semicircle (A) = \\(\\frac{1}{2}\\)\u03c0r\u00b2
\nSo,
\nA = \\(\\frac{1}{2}\\) \u00d7 3.14 \u00d7 \\(\\frac{1,600}{4}\\)
\n= 628 m\u00b2
\nSo,
\nThe area enclosed bythe given figure (A) = 1,600 + 628
\n= 2,228 m\u00b2
\nHence, from the above,
\nWe can conclude that the area enclosed by the given figure is: 2,228 m\u00b2<\/p>\n

Assessment Practice<\/strong>
\nQuestion 17.
\nJulia’s bedroom is 10 feet by 10 feet. She wants to place a circular rug in the corner of her room.
\nPART A
\nShe places a rug with a radius of 2 feet in her room. How much of her bedroom floor, in square feet, is not covered by the rug? Use 3.14 for \u03c0. Round to the nearest tenth.
\nAnswer:
\nIt is given that
\nJulia’s bedroom is 10 feet by 10 feet. She wants to place a circular rug in the corner of her room.
\nSo,
\nThe diameter of the circular rug placed by Julia is: 10 feet
\nNow,
\nWe know that,
\nRadius (r) = \\(\\frac{Diameter}{2}\\)
\nr = \\(\\frac{10}{2}\\)
\nr = 5 feet
\nSo,
\nThe space of bedroom floor that is not covered by the rug (C) = The space of the bedroom floor that is covered by the rug when the radius is 5 meters – The space of the bedroom floor that is covered by the rug when the radius is 2 meters
\nSo,
\nC = 2\u03c0 (5 – 2)
\n= 2 \u00d7 3.14 \u00d7 3
\n= 18.84 feet
\nHence, from the above,
\nWe can conclude that the space of bedroom floor that is not covered by the rug to the neares tenth is: 18.8 feet<\/p>\n

PART B
\nJulia decides she wants a rug that covers about 50% of her floor. Which rug should she buy?
\nA. A rug with a radius of 5 feet
\nB. A rug with a diameter of 5 feet
\nC. A rug with a radius of 4 feet
\nD. A rug with a diameter of 4 feet
\nAnswer:
\nIt is given that Julia’s bedroom is 10 feet by 10 feet
\nSo,
\nThe diameter of the circular rug is: 10 feet
\nNow,
\nThe rug covers the whole bedroom floor when the diameter is 10 feet
\nSo,
\nWhen Julia decides to cover the bedroom floor only 50% with the rug,
\nThen, we have to find only the radius of the circular rug i.e., \\(\\frac{10}{2}\\)
\nHence, from the above,
\nWe can conclude that option A matches the given situation
\n\"\"<\/p>\n

Question 18.
\nThe circumference of a hubcap of a tire is 81.58 centimeters. Find the area, in square centimeters, of this hubcap. Use 3.14 as an approximation for \u03c0. Round your answer to the nearest whole centimeter.
\nAnswer:
\nIt is given that
\nThe circumference of a hubcap of a tire is 81.58 centimeters.
\nNow,
\nWe know that,
\nThe circumference of a circle (C) = 2\u03c0r
\nSo,
\n81.58 = 2 \u00d7 3.14 \u00d7 r
\nr = \\(\\frac{81.58}{6.28}\\)
\nr = 12.99 cm
\nNow,
\nThe area of a circle (A) = \u03c0r\u00b2
\nSo,
\nA = 3.14 \u00d7 (12.99)\u00b2
\n= 529.8 cm\u00b2
\n\u2248 530 cm\u00b2
\nHence, from the above,
\nWe can conclude that the area of the circular hubcap is about 530 cm\u00b2<\/p>\n

3-Act Mathematical Modeling: Whole Lotta Dough<\/h3>\n

\u0410\u0421\u0422 1<\/strong>
\nQuestion 1.
\nAfter watching the video, what is the first question that comes to mind?
\nAnswer:<\/p>\n

Question 2.
\nWrite the Main Question you will answer.
\nAnswer:<\/p>\n

Question 3.
\nConstruct Arguments Predict an answer to this Main Question. Explain your prediction.
\n\"Envision
\nAnswer:<\/p>\n

Question 4.
\nOn the number line below, write a number that is too small to be the answer. Write a number that is too large.
\n\"Envision
\nAnswer:<\/p>\n

Question 5.
\nPlot your prediction on the same number line.
\nAnswer:<\/p>\n

ACT 2<\/strong>
\nQuestion 6.
\nWhat information in this situation would be helpful to know? How would you use that information?
\n\"Envision
\nAnswer:<\/p>\n

Question 7.
\nUse Appropriate Tools What tools can you use to solve the problem? Explain how you would use them strategically.
\nAnswer:<\/p>\n

Question 8.
\nModel with Math Represent the situation using mathematics. Use your representation to answer the Main Question.
\nAnswer:<\/p>\n

Question 9.
\nWhat is your answer to the Main Question? Is it higher or lower than your prediction? Explain why.
\n\"Envision
\nAnswer:<\/p>\n

ACT 3<\/strong>
\nQuestion 10.
\nWrite the answer you saw in the video.
\n\"Envision
\nAnswer:<\/p>\n

Question 11.
\nReasoning Does your answer match the answer in the video? If not, what are some reasons that would explain the difference?
\nAnswer:<\/p>\n

Question 12.
\nMake Sense and Persevere Would you change your model now that you know the answer? Explain.
\n\"Envision
\nAnswer:<\/p>\n

ACT 3 Extension<\/strong>
\nQuestion 13.
\nModel with Math Explain how you used a mathematical model to represent the situation. How did the model help you answer the Main Question?
\nAnswer:<\/p>\n

Question 14.
\nReasoning Explain why your answer to the Main Question does not involve the symbol \u03c0.
\nAnswer:<\/p>\n

SEQUEL<\/strong>
\nQuestion 15.
\nUse Structure If the regular pizza costs $8.99, how much do you think the big pizza costs?
\n\"Envision
\nAnswer:<\/p>\n

Lesson 8.7 Describe Cross Sections<\/h3>\n

Solve & Discuss It!<\/strong><\/p>\n

How could Mrs. Mendoza divide the ream of paper equally between two art classes? She has a paper cutter to slice the paper if needed. What will the dimensions for each sheet of paper be once she has divided the ream? How many sheets will each class receive?
\n\"Envision<\/p>\n

I can… determine what the cross-section looks like when a 3D figure is sliced.
\nAnswer:
\nIt is given that
\nMrs. Mendoza has a paper cutter to slice the paper.
\nNow,
\nThe given figure is:
\n\"Envision
\nNow,
\nFrom the given figure,
\nWe can observe that,
\nA ream consists of 500 sheets
\nWe can also observe that
\nThe dimensions of the ream are:
\nLength of the ream: 8\\(\\frac{1}{2}\\) inches
\nWidth of the ream: 11 inches
\nNow,
\nFrom the given figure,
\nWe can observe that
\nThe shape of the ream is: Rectangle
\nNow,
\nWe know that,
\nEven we cut a ream of paper, each sheet of paper has the same dimensions as a ream of paper
\nSo,
\nThe dimensions of each sheet of paper after Mrs. Mendoza divided the ream into 2 equal parts are:
\nLength: 8\\(\\frac{1}{2}\\) inches
\nWidth: 11 inches
\nNow,
\nThe number of sheets of paper each class will get after dividing the ream = \\(\\frac{500}{2}\\)
\n= 250 sheets
\nHence, from the above,
\nWe can conclude that
\nThe number of sheets each class will receive is: 250
\nThe dimensions of each sheet of paper once Mrs. Mendoza divided the ream is:
\nLength: 8\\(\\frac{1}{2}\\) inches
\nWidth: 11 inches<\/p>\n

Focus on math practices
\nUse Structure How would the number of sheets of paper each class receives change if Mrs. Mendoza started with 300 sheets?
\nAnswer:
\nIt is given that
\nMrs. Mendoza has 500 sheets of paper and she divided the sheets equally between the two classes
\nNow,
\nIf Mrs. Mendoza started with 300 sheets of paper, then
\nThe number of sheets divided by Mrs. Mendoza equally between the two classes = \\(\\frac{300}{2}\\)
\n= 150
\nHence, from the above,
\nWe can conclude that the number of sheets each class receives when Mrs. Mendoza started with 300 sheets is: 150<\/p>\n

Essential Question
\n<\/strong> How do the faces of a three-dimensional figure determine the two-dimensional shapes created by slicing the figure?
\nAnswer:
\nA cross-section is a new face you see when you slice through a three-dimensional figure.
\nExample:
\nIf you slice a rectangular pyramid parallel to the base, you get a smaller rectangle as the cross-section.
\nIf you slice a rectangular pyramid parallel to the apex, you get a smaller triangle as the cross-section.<\/p>\n

Try It!<\/strong>
\nZachary made a vertical slice that was parallel to the left and right faces of a bread roll. What shape is the cross-section, and what are its dimensions?
\n\"Envision
\nThe shape of the cross section is a __________ that is __________ inches by ________ inches.
\nAnswer:
\nIt is given that
\nZachary made a vertical slice that was parallel to the left and right faces of a bread roll
\nNow,
\nThe given figure is:
\n\"Envision
\nNow,
\nFrom the given figure,
\nWe can observe that
\nWhen Zachary made a vertical slice that was parallel to the left and right faces of a bread roll,
\nThe shape of the cross-section Zachary can get is: Rectangle
\nNow,
\nThe dimensions of the rectangle that we obtained by slicing is:
\nLength: 3 in.
\nWidth: 2 in.
\nHence, from the above,
\nWe can conclude that
\nThe shape of the cross-section is a “Rectangle” that is 3 inches \u00d7 2 inches.<\/p>\n

Convince Me! What are the shapes of horizontal and vertical cross-sections of a rectangular prism, and how can you determine the dimensions of the cross-sections?
\nAnswer:
\nThe horizontal cross-section of a rectangular prism is: Rectangle
\nThe vertical cross-section of a rectangular prism is: Rectangle
\nNow,
\nThe dimensions of the cross-sections are:
\na. One dimension of the cross-section will be the height of the prism.
\nb. Another dimension of the cross-section will be the length of the rectangle.<\/p>\n

Try It!<\/strong>
\nDraw the cross-section that is created when a vertical plane intersects the top vertex and the shorter edge of the base of the pyramid shown. What is the area of the cross section?
\n\"Envision
\nAnswer:
\nThe given figure is:
\n\"Envision
\nNow,
\nFrom the given figure,
\nWe can observe that
\nThe cross-section that is created when a vertical plane intersects the top vertex and the shorter edge of the base of the pyramid is: Triangle
\nNow,
\nWe know that,
\nThe area of a triangle (A) = \\(\\frac{1}{2}\\) \u00d7 Base \u00d7 Height
\nSo,
\nA = \\(\\frac{1}{2}\\) \u00d7 7 \u00d7 4
\n= 14 in.\u00b2
\nHence, from the above,
\nWe can conclude that the area of the cross-section is: 14 in.\u00b2<\/p>\n

KEY CONCEPT<\/strong><\/p>\n

A cross section is the two-dimensional shape exposed when a three-dimensional figure is sliced. The shape and dimensions of a cross section in a rectangular prism are the same as the faces that are parallel to the slice.
\n\"Envision<\/p>\n

Do You Understand?<\/strong><\/p>\n

Question 1.
\nEssential Question<\/strong> How do the faces of a three-dimensional figure determine the two-dimensional shapes created by slicing the figure?
\nAnswer:
\nA cross-section is a new face you see when you slice through a three-dimensional figure.
\nExample:
\nIf you slice a rectangular pyramid parallel to the base, you get a smaller rectangle as the cross-section.
\nIf you slice a rectangular pyramid parallel to the apex, you get a smaller triangle as the cross-section.<\/p>\n

Question 2.
\nGeneralize What are the shapes of the cross-sections that are parallel or perpendicular to the bases of a right rectangular prism?
\nAnswer:
\nThe shape of the cross-section that is parallel to the base of a right rectangular prism is: Rectangle
\nThe shape of the cross-section that is perpendicular to the base of a right rectangular prism is: Triangle<\/p>\n

Question 3.
\nGeneralize What are the shapes of the horizontal cross-sections of a right rectangular pyramid? What are the shapes of vertical cross-sections through the vertex opposite the base?
\nAnswer:
\nThe shape of the horizontal cross-section of a right rectangular pyramid is: Rectangle
\nThe shape of the vertical cross-section through the vertex that is opposite to the base is: Triangle<\/p>\n

Do You Know How?<\/strong><\/p>\n

Question 4.
\nThe divider in a desk drawer is a cross-section that is parallel to the front of the drawer. What is its shape, and what are its dimensions?
\n\"Envision
\nAnswer:
\nIt is given that
\nThe divider in a desk drawer is a cross-section that is parallel to the front of the drawer
\nNow,
\nThe given figure is:
\n\"Envision
\nNow,
\nFrom the given figure,
\nWe can observe that
\nThe shape of the cross-section of the divider that is parallel in front of the door is: Rectangle
\nNow,
\nThe dimensions of the cross-section of the divider are:
\nLength: 8 cm
\nWidth: 5 cm
\nHence, from the above,
\nWe can conclude that
\nThe shape of the cross-section of the divider is: Rectangle
\nThe dimensions of the cross-section of the divider is:
\nLength: 8 cm
\nWidth: 5 cm<\/p>\n

Question 5.
\nUse the diagram to answer the questions.
\n\"Envision
\na. Draw the cross-section that is formed when the pyramid is sliced vertically through its vertex and its right face, perpendicular to its base.
\nAnswer:
\nThe given figure is:
\n\"Envision
\nHence,
\nThe cross-section that is formed when the pyramid is sliced vertically through its vertex and its right face that is perpendicular to its base is:
\n\"\"<\/p>\n

b. What is the area of this cross section?
\nAnswer:
\nFrom part (a),
\nWe can observe hat
\nThe cross-section that is formed when the pyramid is sliced vertically through its vertex and its right face that is perpendicular to its base is:
\n\"\"
\nNow,
\nWe know that,
\nThe area of a triangle (A) = \\(\\frac{1}{2}\\) \u00d7 Base \u00d7 Height
\nSo,
\nThe area of the given cross-section (A) = \\(\\frac{1}{2}\\) \u00d7 9 \u00d7 10
\n= 45 mm\u00b2
\nHence, from the above,
\nWe can conclude that the area of the given cross-section is: 45 mm\u00b2<\/p>\n

Question 6.
\nWhat are the dimensions of the vertical cross section shown on this right rectangular prism?
\n\"Envision
\nAnswer:
\nThe given figure is:
\n\"Envision
\nNow,
\nFrom the given figure,
\nWe can observe that,
\nThe vertical cross-section of the given right rectangular prism is: Rectangle
\nNow,
\nThe dimensions of the vertical cross-section of the given right rectangular prism is:
\nLength: 5 in.
\nWidth: 4 in.
\nHence, from the above,
\nWe can conclude that
\nThe dimensions of the vertical cross-section shown on the given right rectangular prism is:
\nLength: 5 in.
\nWidth: 4 in.<\/p>\n

Question 7.
\nBe Precise Describe the cross section that is formed by a vertical plane, perpendicular to the base of the pyramid, that intersects the 9-in. edge and the top vertex of the pyramid shown.
\n\"Envision
\nAnswer:
\nThe given figure is:
\n\"Envision
\nHence,
\nThe cross-section that is formed by a vertical plane, perpendicular to the base of the pyramid, that intersects the 9-in. edge and the top vertex of the given pyramid is:
\n\"\"<\/p>\n

Question 8.
\nMason is slicing butter for the meal he is preparing. Describe the vertical cross section when the knife slices through the butter, parallel to its sides.
\n\"Envision
\nAnswer:
\nIt is given that
\nMason is slicing butter for the meal he is preparing
\nNow,
\nThe given figure is:
\n\"Envision
\nHence,
\nThe vertical cross-section when the knife slices through the butter, parallel to its sides is:
\n\"\"<\/p>\n

Question 9.
\na. Look for Relationships What are the dimensions of the vertical cross section?
\n\"Envision
\nAnswer:
\nThe given figure is:
\n\"Envision
\nHence,
\nThe dimensions of the vertical cross-section of the given figure is:
\n\"\"<\/p>\n

b. What would be the dimensions of a horizontal cross section?
\nAnswer:
\nThe given figure is:
\n\"Envision
\nHence,
\nThe horizontal cross-section of the given figure is:
\n\"\"<\/p>\n

Question 10.
\nUse the figure to the right.
\na. Describe the cross section shown.
\n\"Envision
\nAnswer:
\nThe given figure is:
\n\"Envision
\nNow,
\nFrom the given figure,
\nWe can observe that
\nThe horizontal cross-section of the given figure is: Rectangle
\nThe vertical cross-section of the given figure is: Rectangle
\nHence,
\nThe dimensions of the horizontal cross-section of the given figure is:
\nLength: 4 ft
\nWidth: 6 ft
\nThe dimensions of the vertical cross-section of the given figure is:
\nLength: 11 ft
\nWidth: 6 ft<\/p>\n

b. Is it possible to have a horizontal cross section with different dimensions if you had the plane intersect the prism at another height? Explain.
\nAnswer:
\nYes, it is possible to have a horizontal cross section with different dimensions if you had the plane intersect the prism at another height
\nNow,
\nThe given figure is:
\n\"Envision
\nNow,
\nFrom the given figure,
\nWe can observe that
\nIf we make a horizontal cross-section when another plane intersects the prism at another height, then
\nThe dimensions of the new horizontal cross-section is:
\nLength: 11 ft
\nWidth: 6 ft<\/p>\n

Question 11.
\nMake Sense and Persevere The base of a right rectangular pyramid has a length of 12 centimeters, a width of 6 centimeters, and a height of 14 centimeters. Describe the cross-section formed by a horizontal plane that intersects the faces of the pyramid above the base.
\nAnswer:
\nIt is given that
\nThe base of a right rectangular pyramid has a length of 12 centimeters, a width of 6 centimeters, and a height of 14 centimeters
\nNow,
\nWe know that,
\nThe horizontal cross-section of a right rectangular pyramid is: Rectangle
\nThe vertical cross-section of a right rectangular pyramid is: Triangle
\nSo,
\nThe dimensions of the horizontal cross-section of the given right rectangular pyramid is:
\nLength: 12 cm
\nWidth: 6 cm
\nHence,
\nThe representation of the horizontal cross-section of the given right rectangular pyramid is:
\n\"\"<\/p>\n

Question 12.
\nHigher Order Thinking Luis makes blocks from a painted piece of wood with dimensions of 27 inches \u00d7 24 inches \u00d7 1.5 inches. To make 72 blocks, the wood is cut into 3-inch squares. Draw two pictures showing the horizontal cross section and the vertical cross section of each block.
\nAnswer:
\nIt is given that
\nLuis makes blocks from a painted piece of wood with dimensions of 27 inches \u00d7 24 inches \u00d7 1.5 inches. To make 72 blocks, the wood is cut into 3-inch squares.
\nSo,
\nThe dimensions of each block is:
\nLength: \\(\\frac{27}{3}\\) = 9 inches
\nWidth: \\(\\frac{24}{3}\\) = 8 inches
\nHeight: \\(\\frac{1.5}{3}\\) = 0.5 inches
\nNow,
\nIt is given that the 72 blocks are cut into 3-inch squares
\nSo,
\nThe figure made by the given dimensions is: Cuboid
\nNow,
\nThe vertical cross-section of the given cuboid is: Rectangle
\nThe horizontal cross-section of the given cuboid is: Rectangle
\nHence,
\nThe representation of the vertical and horizontal cross-sections of the cuboid are:
\n\"\"<\/p>\n

Question 13.
\nMake Sense and Persevere The area of the cross section shown is 52 square yards. What is the length of the unknown side of the base of the pyramid?
\n\"Envision
\nAnswer:
\nIt is given that
\nThe area of the cross-section shown is 52 square yards.
\nNow,
\nThe given figure is:
\n\"Envision
\nNow,
\nFrom the above,
\nWe can observe that the given figure is: Right Rectangular pyramid
\nNow,
\nWe know that,
\nThe vertical cross-section of the right rectangular pyramid is: Triangle
\nNow,
\nWe know that,
\nThe area of a triangle (A) = \\(\\frac{1}{2}\\) \u00d7 Base \u00d7 Height
\nSo,
\n52 = \\(\\frac{1}{2}\\) \u00d7 x \u00d7 13
\nx = \\(\\frac{52 \u00d7 2}{13}\\)
\nx = 8 yd
\nHence, from the above,
\nWe can conclude that the length of the unknown side of the base of the given pyramid is: 8 yd<\/p>\n

Question 14.
\nA waiter slices a cake shaped like a square pyramid vertically through the top point.
\na. Make Sense and Persevere Draw the cross section that is made by slicing the cake in this way.
\n\"Envision
\nAnswer:
\nIt is given that
\nA waiter slices a cake shaped like a square pyramid vertically through the top point.
\nNow,
\nThe given figure is:
\n\"Envision
\nNow,
\nFrom the given figure,
\nWe can observe that
\nIf we made the vertical cross-section of the given square pyramid, then
\nWe will get the vertical cross-section as a Right triangle
\nHence,
\nThe representation of the vertical cross-section of the given square pyramid is:
\n\"\"<\/p>\n

b. What is the area of this cross section?
\nAnswer:
\nFrom part (a),
\nWe can observe that
\nThe representation of the vertical cross-section of the given square pyramid is:
\n\"\"
\nNow,
\nWe know that,
\nIn a right triangle,
\n(Hypotenuse)\u00b2 = (Base)\u00b2 + (Side)\u00b2
\nSo,
\nIn the given triangle,
\n18\u00b2 = 14\u00b2 + h\u00b2
\nh\u00b2 = 18\u00b2 – 14\u00b2
\nh = 11.3 inches
\nNow,
\nWe know that,
\nThe area of a triangle (A) = \\(\\frac{1}{2}\\) \u00d7 Base \u00d7 Height
\nSo,
\nA = \\(\\frac{1}{2}\\) \u00d7 14 \u00d7 11.3
\n= 79.19 yd\u00b2
\nHence, from the above,
\nWe can conclude that the area of the given cross-section is: 79.19 yd\u00b2<\/p>\n

Question 15.
\nMiranda says that the triangle below represents the cross section of the rectangular pyramid shown.
\n\"Envision
\nWhat mistake might Miranda have made?
\nAnswer:
\nIt is given that
\nMiranda says that the triangle below represents the cross-section of the rectangular pyramid shown.
\nNow,
\nThe given figure is:
\n\"Envision
\nNow,
\nWe know that,
\nIn a rectangular pyramid,
\nThe vertical cross-section is: Triangle
\nThe horizontal cross-section is: Rectangle
\nSo,
\nAccording to the given information,
\nMiranda did not mention what type of cross-section she had drawn
\nHence, from the above,
\nWe can conclude that the mistake made by Miranda is:
\nShe did not mention what type of cross-section she made<\/p>\n

Assessment Practice<\/strong>
\nQuestion 16.
\nEstimate, to the nearest whole number, the number of vertical cross sections needed to equal the area of the base of the figure to the right. Explain how you made your estimate, and decide whether your estimate is higher or lower than the actual number.
\n\"Envision
\nAnswer:
\nThe given figure is:
\n\"Envision<\/p>\n

Now,
\nFrom the given figure,
\nWe can observe that the given 3-d shape is: Cuboid
\nNow,
\nWe know that,
\nThe vertical cross-section of a cuboid is: Rectangle
\nThe horizontal cross-section of a cuboid is: Rectangle
\nNow,
\nFrom the given figure,
\nThe dimensions of a horizontal cross-section of the given figure is:
\nLength: 8 cm
\nWidth: 6 cm
\nThe dimensions of a vertical cross-section of the given figure is:
\nLength: 6 cm
\nWidth: 3 cm
\nNow,
\nWe know that,
\nThe area of a rectangle = Length \u00d7 Width
\nSo,
\nThe area of a horizontal cross-section of the given figure is:
\nA = 8 \u00d7 6 = 48 cm\u00b2
\nThe area of a vertical cross-section of the given figure is:
\nA = 6 \u00d7 3 = 18 cm\u00b2
\nHence, from the above,
\nWe can conclude that
\nThe number of vertical cross-sections < The area of the base of the given figure<\/p>\n

Lesson 8.8 Solve Problems Involving Surface Area<\/h3>\n

Solve & Discuss It!<\/strong><\/p>\n

Alaya will paint the outside of a box in three different colors. Decide how she could paint the box. What is the total area that each color will cover?
\n\"Envision
\nI can… find the area and surface area of 2-dimensional composite shapes and 3-dimensional prisms.
\nAnswer:
\nIt is given that
\nAlaya will paint the outside of a box in three different colors
\nNow,
\nThe given figure is:
\n\"Envision
\nNow,
\nFrom the given figure,
\nWe can observe that the given figure is a “Cuboid” which has 6 faces
\nNow,
\nThe different ways that Alaya can paint are:
\na. She can paint all the faces of the box with the same color
\nb. She can paint the faces of the box with alternative colors i.e., 1 face is colored with 1 color, 2nd face is colored with 1 color, etc.
\nNow,
\nWe know that,
\nThe total surface area of a cuboid (A) = 2 (lw + wh + lh)
\nWhere,
\n‘l’ is the length of the cuboid
\n‘w’ is the width of the cuboid
\n‘h’ is the height of the cuboid
\nNow,
\nFrom the given figure,
\nThe length of the cuboid (l) is 32 in.
\nThe width of the cuboid (w) is 16 in.
\nThe height of the cuboid (h) is 14 in.
\nSo,
\nA = 2 (32 \u00d7 16 + 16 \u00d7 14 + 32 \u00d7 14)
\n= 2 (512 + 224 + 448)
\n= 2,368 in.\u00b2
\nSo,
\nThe total surface area that each color will cover = \\(\\frac{2,368}{6}\\)
\n= 394.66 in.\u00b2
\nHence, from the above,
\nWe can conclude that
\nThe different ways that Alaya can paint are:
\na. She can paint all the faces of the box with the same color
\nb. She can paint the faces of the box with alternative colors i.e., 1 face is colored with 1 color, 2nd face is colored with 1 color, etc.
\nThe total area covered by each color is: 394.66 in.\u00b2<\/p>\n

Make Sense and Persevere
\nWhat do you know about the faces of a rectangular prism?
\nAnswer:
\nThe facts about the rectangular prism are:
\na. A rectangular prism is a polyhedron with two congruent and parallel bases.
\nb. Because of its cross-section along the length, it is said to be a prism.
\nc. A rectangular prism has 6 faces, 8 vertices, and 12 edges. Its base and top are always rectangles.
\nd. It has 3 dimensions They are length, width, and height.<\/p>\n

Focus on math practices
\nReasoning Trista paints each pair of opposite sides of the box with the same color. How many different areas does she need to find to determine the total area covered by each color? Explain.
\nAnswer:
\nThe given figure is:
\n\"Envision
\nNow,
\nIt is given that
\nTrista paints each pair of opposite sides of the box with the same color
\nSo,
\nEach color covers two opposite pairs of rectangles.
\nSo,
\nWith the help of each color,
\nWe can determine the total area of the two opposite sides of the box
\nHence, from the above,
\nWe can conclude that
\nshe needs to find 3 different areas to determine the total area covered by each color<\/p>\n

Essential Question<\/strong>
\nHow is finding the area of composite two-dimensional figures similar to finding the surface area of three-dimensional figures?
\nAnswer:
\nTo find the area of something, you start by multiplying the two dimensions together. To find the area of a rectangle, you multiply the length by the width. Although the area is a two-dimensional measurement, it can also be used with three-dimensional objects.<\/p>\n

Try It!<\/strong><\/p>\n

This diagram shows the area of a room to be carpeted. What will be the area of the new carpet?
\n\"Envision
\nA = ________ = ________ ft2<\/sup> B = _______ = _______ ft2<\/sup>
\nTotal area = ______ + ______
\nThe area of the new carpet is ________ square feet
\nAnswer:
\nIt is given that
\nThe below diagram shows the area of a room to be carpeted.
\nNow,
\nThe given figure is:
\n\"Envision
\nNow,
\nFrom the given figure,
\nWe can observe that
\nThe total area of a room that is carpeted = The area of a rectangle + The area of a triangle
\nNow,
\nFrom the given figure,
\nThe dimensions of the rectangle are:
\nLength: 6 feet
\nWidth: 5 feet
\nThe dimensions of the triangle are:
\nHeight: 6 feet
\nBase: 4 feet
\nNow,
\nWe know that,
\nThe area of a rectangle = Length \u00d7 Width
\n= 6 \u00d7 5
\n= 30 feet\u00b2
\nNow,
\nWe know that,
\nThe area of a triangle = \\(\\frac{1}{2}\\) \u00d7 Base \u00d7 height
\n= \\(\\frac{1}{2}\\) \u00d7 6 \u00d7 4
\n= 12 feet\u00b2
\nSo,
\nThe area of the new carpet = 30 + 12
\n= 42 feet\u00b2
\nHence, from the above,
\nWe can conclude that the area of the new carpet is: 42 feet\u00b2<\/p>\n

Convince Me! How does knowing the area of familiar shapes help find the total area of a composite shape?
\nAnswer:
\nA composite figure is made up of simple geometric shapes. To find the area of a composite figure or other irregular-shaped figures, divide it into simple, non-overlapping figures. Find the area of each simpler figure, and then add the areas together to find the total area of the composite\u00a0figure<\/p>\n

Try It!<\/strong><\/p>\n

Hiromi is painting the front and back of a barn. Each can of paint covers 32 square feet. How many cans of paint does Hiromi need to cover the entire front and back of the barn?
\n\"Envision
\nAnswer:
\nIt is given that
\nHiromi is painting the front and back of a barn. Each can of paint covers 32 square feet.
\nNow,
\nThe given figure is:
\n\"Envision
\nNow,
\nFrom the given figure,
\nWe can observe that
\nThe total area of the barn (A) = The area of the front + The area of the back
\n= 2 (The area of the front) (or) 2 (The area of the back)
\nNow,
\nThe front side of the barn is made up of a rectangle and a triangle
\nNow,
\nThe area of the front side of the barn = The area of the rectangle + The area of the triangle
\n= Length \u00d7 Width + \\(\\frac{1}{2}\\) \u00d7 Base \u00d7 Height
\n= 11 \u00d7 12 + \\(\\frac{1}{2}\\) \u00d7 12 \u00d7 (15 – 11)
\n= 132 + 24
\n= 156 ft\u00b2
\nSo,
\nThe total area of the barn (A) = 2 \u00d7 156
\n= 312 ft\u00b2
\nNow,
\nThe number of cans of paint does Hiromi need to cover the entire front and back of the barn = \\(\\frac{The total area of the barn}{The number of square feet each can covers}\\)
\n= \\(\\frac{312}{32}\\)
\n= 9.75
\n\u2245 10 cans of paint
\nHence, from the above,
\nWe can conclude that Hiromi needs to cover 10 cans of paint to cover the entire front and back of the barn<\/p>\n

KEY CONCEPT<\/strong><\/p>\n

The area of a two-dimensional composite figure is the sum of the areas of all the shapes that compose it. The surface area of a three-dimensional composite figure is the sum of the areas of all its faces.
\nTwo-dimensional composite figure
\n\"Envision<\/p>\n

Three-dimensional composite figure
\n\"Envision
\nSurface area of shape A + Surface area of shape B = Surface area of composite shape<\/p>\n

Do You Understand?<\/strong><\/p>\n

Question 1.
\nEssential Question<\/strong> How is finding the area of composite two-dimensional figures similar to finding the surface area of three-dimensional figures?
\nAnswer:
\nTo find the area of something, you start by multiplying the two dimensions together. To find the area of a rectangle, you multiply the length by the width. Although the area is a two-dimensional measurement, it can also be used with three-dimensional objects.<\/p>\n

Question 2.
\nMake Sense and Persevere Laine wants to determine the amount of fabric needed to cover a triangular prism-shaped box. She begins by measuring the dimensions of the box. Explain her next steps.
\nAnswer:
\nIt is given that
\nLaine wants to determine the amount of fabric needed to cover a triangular prism-shaped box. She begins by measuring the dimensions of the box
\nNow,
\nThe next steps she has to follow are:
\nStep 1:
\nDivide the given triangular prism into the number of parts that have many faces as the given triangular prism has.
\nWe know that,
\nThe triangular prism has: 5 faces
\nSo,
\nThe number of smaller triangles are: 5
\nStep 2:
\nFind the area of each triangle
\nWe know that,
\nThe area of a triangle (A) = \\(\\frac{1}{2}\\) \u00d7 Base \u00d7 Height
\nStep 3:
\nAdd the areas of the 5 triangles to find the area of the given triangular prism
\nHence, from the above,
\nWe can conclude that the amount of fabric needed to cover a triangular prism is its total area<\/p>\n

Question 3.
\nUse Structure Explain how you would find the surface area of the figure below.
\n\"Envision
\nAnswer:
\nThe given figure is:
\n\"Envision
\nNow,
\nFrom the given figure,
\nWe can observe that the given figure is made up of:
\na. Rectangle b. Pentagon
\nSo,
\nThe total surface area of the given figure = Total area of the rectangle + Total area of the pentagon
\nNow,
\nFrom the given figure,
\nWe can observe that
\nThere are:
\na. 5 Rectangles b. 2 Pentagons
\nHence,
\nThe total surface area of the given figure (A) = The area of 5 Rectangles + The area of 2 Pentagons<\/p>\n

Do You Know How?<\/p>\n

Question 4.
\nPaula is painting a henhouse. If a can of paint will cover 24 square feet, how many cans of paint does she need to buy? Explain the steps she might take to solve this problem.
\nAnswer:
\nIt is given that Paula is painting a henhouse
\nNow,
\nThe representation of a henhouse is:
\n\"\"
\nNow,
\nFrom the given figure,
\nWe can observe that
\nThe henhouse is made up of:
\na. 5 Rectangles b. 4 Triangles
\nNow,
\nThe steps that Paula might take to find the number of cans she needed to buy are:
\nStep 1:
\nDivide the henhouse into simple 2-d figures i.e, into rectangles and triangles
\nStep 2:
\nFind the areas of the 2-d figures
\nStep 3:
\nFind the total area of the henhouse by adding all the areas of 2-d figures
\nStep 4:
\nTo find the number of cans Paula needed to buy,
\nThe number of cans Paula needed to buy = \\(\\frac{The total area of the henhouse}{The area covered by 1 can of paint}\\)
\nHence, from the above,
\nWe can conclude that
\nBy following the above steps, Paula can solve her problem<\/p>\n

Question 5.
\nFind the area of the composite figure. The two triangles have the same dimensions.
\n\"Envision
\nAnswer:
\nThe given figure is:
\n\"Envision
\nNow,
\nFrom the above figure,
\nWe can observe that
\nThe area of the composite figure (A) = The area of the 2 rectangles + The area of the 2 triangles
\nNow,
\nWe know that,
\nThe area of a rectangle (A) = Length \u00d7 Width
\nThe area of a triangle (A) = \\(\\frac{1}{2}\\) \u00d7 Base \u00d7 Height
\nSo,
\nA = (7.5 \u00d7 13) + (7.5 \u00d7 6) + 2 \u00d7 \\(\\frac{1}{2}\\) \u00d7 2.5 \u00d7 6
\n= 97.5 + 45 + 15
\n= 157.5 cm\u00b2
\nHence, from the above,
\nWe can conclude that the area of the composite figure is: 157.5 cm\u00b2<\/p>\n

Question 6.
\nA stage block is being covered in carpet. The dimensions of the block are 2 feet by 3 feet by 6 feet. Every surface will need covering except for the surface touching the floor, which is 3 feet by 6 feet. How would you calculate the surface area that needs covering?
\nAnswer:
\nIt is given that
\nA stage block is being covered in carpet. The dimensions of the block are 2 feet by 3 feet by 6 feet. Every surface will need covering except for the surface touching the floor, which is 3 feet by 6 feet.
\nNow,
\nThe surface area that needs covering = The total surface area – The surface area of the floor
\nNow,
\nFrom the given information,
\nWe can say that the carpet is in the form of a rectangular prism
\nNow,
\nWe know that,
\nThe surface area of a rectangular prism (A) = 2 (lw + wh + lh)
\nWhere,
\n‘l’ is the length
\n‘w’ is the width
\n‘h’ is the height
\nSo,
\nThe total surface area (A) = 2 (2 \u00d7 3 + 3 \u00d7 6 + 2 \u00d7 6)
\n= 2 (6 + 18 + 12)
\n= 72 feet\u00b2
\nNow,
\nFrom the given information,
\nWe can observe that the floor is in the form of a rectangle
\nSo,
\nThe surface area of the floor (A) = 3 \u00d7 6
\n= 18 ft\u00b2
\nSo,
\nThe surface area that needs covering = The total surface area – The surface area of the floor
\n= 72 – 18
\n= 54 feet\u00b2
\nHence, from the above,
\nWe can conclude that the surface area that needs covering is: 54 feet\u00b2<\/p>\n

Practice & Problem Solving<\/strong><\/p>\n

Leveled Practice In 7, fill in the boxes to solve.<\/p>\n

Question 7.
\nJacob is putting tiles on the sections of his yard labeled A, B, and C. What is the area of the parts that need tiles?
\n\"Envision
\nAnswer:
\nIt is given that
\nJacob is putting tiles on the sections of his yard labeled A, B, and C.
\nNow,
\nFrom the given figure,
\nWe can observe that the tiles are made up of parts named as:
\nA: Rectangle B: Rectangle C: Triangle
\nNow,
\nWe know that,
\nThe area of a rectangle (A) = Length \u00d7 Width
\nThe area of\u00a0 triangle A) = \\(\\frac{1}{2}\\) \u00d7 Base \u00d7 Height
\nSo,
\nThe area of Part A:
\nA = 6 \u00d7 3 = 18 m\u00b2
\nThe area of Part B:
\nA = 3 \u00d7 4.5 = 13.5 m\u00b2
\nThe area of Part C:
\nA = \\(\\frac{1}{2}\\) \u00d7 3 \u00d7 3 = 4.5 m\u00b2
\nSo,
\nThe total area of the tiles = The area of Part A + The area of Part B + The area of Part C
\n= 18 + 13.5 + 4.5
\n= 36 m\u00b2
\nHence, from the above,
\nWe can conclude that the area of the parts that need tiles is: 36 m\u00b2<\/p>\n

Question 8.
\nWhat is the total area of the figure?
\n\"Envision
\nAnswer:
\nThe given figure is:
\n\"Envision
\nNow,
\nFrom the given figure,
\nWe can observe that
\nThe area of the given figure (A) = The area of the rectangles + The area of a triangle
\nNow,
\nWe know that,
\nThe area of a rectangle (A) = Length \u00d7 Width
\nThe area of a triangle (A) = \\(\\frac{1}{2}\\) \u00d7Base \u00d7 Height
\nSo,
\nThe area of the given figure (A) = 4 (4 \u00d7 9) + \\(\\frac{1}{2}\\) \u00d7 9 \u00d7 \\(\\sqrt{9\u00b2 – 5\u00b2}\\)
\n= 144 + 18.70
\n= 162.70 ft\u00b2
\nHence, from the above,
\nWe can conclude that the area of the given figure is: 162.70 ft\u00b2<\/p>\n

Question 9.
\nFind the surface area of the prism.
\n\"Envision
\nAnswer:
\nThe given figure is:
\n\"Envision
\nNow,
\nWe know that,
\nThe surface area of a rectangular prism = 2 (lw + wh + lh)
\nWhere,
\n‘l’ is the length
\n‘w’ is the width
\n‘h’ is the height
\nSo,
\nThe area of the given prism (A) = 2 (15 \u00d7 4 + 4 \u00d7 8 + 15 \u00d7 8)
\n= 2 (60 + 32 + 120)
\n= 424 in.\u00b2
\nHence, from the above,
\nWe can conclude that the surface area of the given prism is: 424 in.\u00b2<\/p>\n

Question 10.
\nFind the surface area of the triangular prism. The base of the prism is an isosceles triangle.
\n\"Envision
\nAnswer:
\nThe given figure is:
\n\"Envision
\nNow,
\nFrom the given figure,
\nWe can observe that
\nFor the given Triangular prism,
\nThere are:
\na. 2 Triangles b. 3 Paralleleograms
\nSo,
\nThe area of the triangular prism = The total area of the triangles + The total area of the parallelograms
\nNow,
\nWe know that,
\nThe area of a triangle (A) = \\(\\frac{1}{2}\\) \u00d7 Base \u00d7 height
\nThe area of a parallelogram (A) = Base \u00d7 Height
\nSo,
\nThe total area of the triangular prism (A) = 2 \u00d7 \\(\\frac{1}{2}\\) \u00d7 14 \u00d7 24 + 3 \u00d7 25 \u00d7 48
\n= 336 + 3,600
\n= 3,936 cm\u00b2
\nHence, from the above,
\nWe can conclude that the area of the given triangular prism is: 3,936 cm\u00b2<\/p>\n

Question 11.
\nA block of wood has the shape of a triangular prism. The bases are right triangles. Find its surface area.
\n\"Envision
\nAnswer:
\nIt is given that
\nA block of wood has the shape of a triangular prism. The bases are right triangles
\nNow,
\nThe given triangular prism is:
\n\"Envision
\nNow,
\nFrom the given triangular prism,
\nWe can observe that
\nThe triangular prism has:
\na. 2 Right triangles b. 3 Parallelograms
\nNow,
\nWe know that,
\nThe area of a triangle (A) = \\(\\frac{1}{2}\\) \u00d7 Base \u00d7 height
\nThe area of a parallelogram (A) = Base \u00d7 Height
\nSo,
\nThe total area of the triangular prism (A) = 2 \u00d7 \\(\\frac{1}{2}\\) \u00d7 1.5 \u00d7 2.5 + 3 \u00d7 2.5 \u00d7 15
\n= 3.75 + 112.5
\n= 116.25 in.\u00b2
\nHence, from the above,
\nWe can conclude that the area of the given triangular prism is: 116.25 in.\u00b2<\/p>\n

Question 12.
\nA box has the shape of a rectangular prism. How much wrapping paper do you need to cover the box?
\n\"Envision
\nAnswer:
\nIt is given that A box has the shape of a rectangular prism
\nNow,
\nThe given rectangular prism is:
\n\"Envision
\nNow,
\nTo find the amount of wrapping paper you needed to cover the box,
\nWe have to find the total area of the iven rectangular prism
\nNow,
\nWe know that,
\nThe surface area of a rectangular prism (A) = Length \u00d7 Width \u00d7 Height
\nSo,
\nA = 16 \u00d7 3 \u00d7 15
\n= 720 in.\u00b2
\nHence, from the above,
\nWe can conclude that
\nThe amount of wrapping paper you needed to cover the box is: 720 in.\u00b2<\/p>\n

Question 13.
\nHigher-OrderThinking Find the surface area of the Rectangular Hexagonal prism. Show your work.
\n\"Envision
\nAnswer:
\nThe given rectangular hexagonal prism is:
\n\"Envision
\nNow,
\nFrom the given figure,
\nWe can observe that
\nThe total area of the rectangular hexagonal prism (A) = The area of the 2 cicles + The area of the 6 rectangles
\nNow,
\nWe know that,
\nThe area of a circle (A) = \u03c0r\u00b2
\nThe area of a rectangle (A) = Length \u00d7 Width
\nSo,
\nA = 2 \u00d7 3.14 \u00d7 (4.3)\u00b2 + 6 \u00d7 14 \u00d7 5
\n= 116.11 + 420
\n= 536.11 cm\u00b2
\nHence, from the above,
\nWe can conclude that the area of the given rectangular hexagonal prism is: 536.11 cm\u00b2<\/p>\n

Question 14.
\nA box has the shape of a rectangular prism with a height of 29 centimeters. If the height is increased by 0.7 centimeter, by how much does the surface area of the box increase?
\n\"Envision
\nAnswer:
\nIt is given that
\nA box has the shape of a rectangular prism with a height of 29 centimeters and the height is increased by 0.7 centimeters
\nNow,
\nThe given rectangular prism is:
\n\"Envision
\nNow,
\nWe know that,
\nThe surface area of a rectangular prism (A) = Length \u00d7 Width \u00d7 Height
\nSo,
\nA = 29 \u00d7 6.3 \u00d7 12
\n= 2,192.4 cm\u00b2
\nNow,
\nIf the height of the given rectangular prism is increased by 0.7 cm, then
\nA = (29 + 0.7) \u00d7 6.3 \u00d7 12
\n= 2,245.32 cm\u00b2
\nSo,
\nThe amount of the surface area increased = 2,245.32 – 2,192.4
\n= 52.92 cm\u00b2
\nHence, from the above,
\nWe acn conclude that\u00a0 the amount of surface area increased when the height of the box increased by 0.7 cm is: 52.92 cm\u00b2<\/p>\n

Question 15.
\nThe base of a prism is an equilateral triangle with an area of 73.2 square centimeters. The area of each lateral face is 104 square centimeters. Riley incorrectly claims that the surface area is 250.4 square centimeters.
\na. What is the correct surface area?
\nAnswer:
\nIt is given that
\nThe base of a prism is an equilateral triangle with an area of 73.2 square centimeters. The area of each lateral face is 104 square centimeters. Riley incorrectly claims that the surface area is 250.4 square centimeters.
\nNow,
\nThe total surface area (A) = The area of the base of the prism + The area of each lateral face
\n= 73.2 + 104
\n= 177.2 square cm
\nHence, from the above,
\nWe can conclude that the correct surface area is: 177.2 square cm<\/p>\n

b. What could have been Riley’s error?
\nAnswer:
\nThe error done by Riley is:
\na. She considered the bases of a prism to be 2 and the lateral face as 1
\nSo,
\nThe area she got is 250.4 square cm instead of 177.2 square cm<\/p>\n

Assessment Practice<\/strong>
\nQuestion 16.
\nThe bottom part of this block is a rectangular prism. The top part is a square pyramid. How much paper, in square centimeters, is needed to cover the block completely?
\n\"Envision
\nAnswer:
\nIt is given that
\nThe bottom part of this block is a rectangular prism. The top part is a square pyramid
\nNow,
\nThe given figure is:
\n\"Envision
\nNow,
\nFrom the above,
\nWe can observe that
\nThe area of the given block (A) = The area of the bottom part of the block + The area of the top part of the block
\n= Length \u00d7 Width \u00d7 Height + 5 \u00d7 \\(\\frac{1}{2}\\) \u00d7 Base \u00d7 Height
\nSo,
\nA = 3 \u00d7 4 \u00d7 4 + 5 \u00d7 \\(\\frac{1}{2}\\) \u00d7 6 \u00d7 4
\n= 48 + 60
\n= 108 cm\u00b2
\nHence, from the above,
\nWe can conclude that
\nThe amount of paper, in square centimeters, is needed to cover the block completely is: 108 cm\u00b2<\/p>\n

Lesson 8.9 Solve Problems Involving Volume<\/h3>\n

Solve & Discuss It!<\/strong><\/p>\n

Volunteers at a food pantry pack boxes of soup into crates. How many boxes of soup will fill each crate? Show your work.
\n\"Envision
\nI can… use the area of the base of a three-dimensional figure to find its volume.
\nAnswer:
\nIt is given that
\nVolunteers at a food pantry pack boxes of soup into crates
\nNow,
\nThe given figure is:
\n\"Envision
\nNow,
\nFrom the given figure,
\nWe can observe that the boxes and crates are in the form of a cuboid
\nNow,
\nWe know that,
\nThe volume of a cuboid = Length \u00d7 Width \u00d7 Height
\nSo,
\nThe volume of the box = 4 \u00d7 2 \u00d7 6
\n= 48 in.\u00b3
\nThe volume of the crate = 12 \u00d7 18 \u00d7 12
\n= 2,592 in.\u00b3
\nSo,
\nThe number of boxes of soup that will fill each crate = \\(\\frac{The volume of a crate}{The volume of a box}\\)
\n= \\(\\frac{2,592}{48}\\)
\n= 54 boxes
\nHence, from the above,
\nWe can conclude that the number of boxes of soup that will fill each crate is: 54 boxes<\/p>\n

Look for Relationships How can you layer the soup boxes to cover the bottom of the crate?
\nAnswer:
\nThe given figure in part (a) is:
\n\"Envision
\nNow,
\nFrom the given figure in part (a),
\nWe can observe that
\nWe have to layer the soup boxes in the horizontal position to cover the bottom of the crate<\/p>\n

Focus on math practices<\/strong>
\nReasoning A supplier donated crates to the food pantry that are 15 inches long, instead of 18 inches long. All other dimensions are the same. What is the greatest number of boxes of soup that will fit in the donated crates? How will the volume of the soup vary from the total volume of the crate?
\nAnswer:
\nIt is given that
\nA supplier donated crates to the food pantry that are 15 inches long, instead of 18 inches long. All other dimensions are the same
\nNow,
\nThe given figure is:
\n\"Envision<\/p>\n

Now,
\nFrom the given figure,
\nWe can observe that the boxes and crates are in the form of a cuboid
\nNow,
\nWe know that,
\nThe volume of a cuboid = Length \u00d7 Width \u00d7 Height
\nSo,
\nThe volume of the box = 4 \u00d7 2 \u00d7 6
\n= 48 in.\u00b3
\nThe volume of the crate = 12 \u00d7 15 \u00d7 12
\n= 2,160 in.\u00b3
\nSo,
\nThe number of boxes of soup that will fill each crate = \\(\\frac{The volume of a crate}{The volume of a box}\\)
\n= \\(\\frac{2,160}{48}\\)
\n= 45 boxes
\nHence, from the above,
\nWe can conclude that the greatest number of boxes of soup that will fit in the donated crates is: 45 boxes<\/p>\n

Essential Question<\/strong>
\nHow does the formula for the volume of a prism help you understand what volume of a prism means?
\nAnswer:
\nThe formula for the volume of a prism is:
\nV=Bh
\nwhere,
\nB is the base area
\nh is the height.
\nNow,
\nThe base of the prism is a rectangle
\nSo,
\nThe area of a rectangle = Length \u00d7 Width
\nHence, from the above,
\nWe can conclude that
\nThe formula for the volume of a prism is:
\nV = Length\u00d7 Width \u00d7 Height<\/p>\n

Try It!<\/strong>
\nWhat is the volume of the triangular prism?
\n\"Envision
\nThe volume of the prism is __________ cubic centimeters.
\nAnswer:
\nThe given figure is:
\n\"\"
\nNow,
\nWe know that,
\nThe volume of a triangular prism (V) = Area of a triangle \u00d7 Height
\nNow,
\nWe know that,
\nThe area of a triangle (A) = \\(\\frac{1}{2}\\) \u00d7 Base of the triangle \u00d7 Height of the triangle
\nSo,
\nThe volume of a triangular prism (V) = \\(\\frac{1}{2}\\) \u00d7 8 \u00d7 6 \u00d7 14
\n= 336 cm\u00b3
\nHence, from the above,
\nWe can conclude that the volume of the given triangular prism is: 336 cm\u00b3<\/p>\n

Convince Me! What is the shape of the base of the figure? What are its dimensions? Explain.
\nAnswer:
\nThe given figure is:
\n\"\"
\nNow,
\nFrom the given triangular prism,
\nWe can observe that the shape of the base of the figure is: Triangle
\nNow,
\nFrom the given figure,
\nThe dimensions of the base of the figure are:
\nThe base of the triangle: 8 cm
\nThe height of the triangle: 6 cm
\nHence, from the above,
\nWe can conclude that the dimensions of the base of the given figure is:
\nBase: 8 cm
\nHeight: 6 cm<\/p>\n

Try It!<\/strong>
\nAmber built a custom terrarium for her plants. What is the volume of the terrarium?
\n\"Envision
\nAnswer:
\nIt is given that
\nAmber built a custom terrarium for her plants.
\nNow,
\nThe given figure is:
\n\"Envision
\nNow,
\nFrom the given figure,
\nWe can observe that
\nThe terrarium is made up of two rectangular prisms
\nNow,
\nWe know that,
\nThe volume of a rectangular prism (V) = Length \u00d7 Width \u00d7 Height
\nNow,
\nThe volume of the first rectangular prism (V) = 30 \u00d7 10 \u00d7 10
\n= 3,000 in.\u00b3
\nThe volume of the second rectangular prism (V) = 40 \u00d7 10 \u00d7 12
\n= 4,800 in.\u00b3
\nSo,
\nThe volume of the custom terrarium (V) = The volume of the first rectangular prism + The volume of the second rectangular prism
\n= 3,000 + 4,800
\n= 7,800 in.\u00b3
\nHence, from the above,
\nWe can conclude that the volume of the custom terrarium that was built by Amber is: 7,800 in.\u00b3<\/p>\n

KEY CONCEPT<\/strong><\/p>\n

You can use formulas to solve problems involving the volume of three-dimensional figures.
\nFind the volume of a composite figure by finding the sum of the volumes of each solid figure.
\n\"Envision
\nYou can use the formula V = Bh to find the volume or unknown dimension of a solid figure.
\n\"Envision<\/p>\n

Do You Understand?<\/strong><\/p>\n

Question 1.
\nEssential Question<\/strong> How does the formula for volume of a prism help you understand what volume of a prism means?
\nAnswer:
\nThe formula for the volume of a prism is:
\nV=Bh
\nwhere,
\nB is the base area
\nh is the height.
\nNow,
\nThe base of the prism is a rectangle
\nSo,
\nThe area of a rectangle = Length \u00d7 Width
\nHence, from the above,
\nWe can conclude that
\nThe formula for the volume of a prism is:
\nV = Length\u00d7 Width \u00d7 Height<\/p>\n

Question 2.
\nLook for Relationships If you know the volume of a three-dimensional figure, how can you find a missing dimension of the figure?
\nAnswer:
\nWe know that,
\nThe volume of any three-dimensional figure is:
\nV = Length \u00d7 Width \u00d7 Height
\nNow,
\nIf we have any missing dimension in the figure that is related to the volume, then we will first multiply the known dimensions that are present with the unknown dimension and then divide the product of the known dimensions with the volume of the three-dimensional figure<\/p>\n

Question 3.
\nMake Sense and Persevere How do you find the volume of a three-dimensional figure that can be decomposed into prisms?
\nAnswer:
\nWe know that,
\nA prism is a three-dimensional figure
\nWe know that,
\nThe volume of a prism (V) = Length \u00d7 Width \u00d7 Height
\nSo,
\nIf we have to find the volume of a three-dimensional figure that can be decomposed into prisms, then add all the volumes of the decomposed prisms<\/p>\n

Do You Know How?<\/strong><\/p>\n

Question 4.
\nAn aquarium has a regular hexagonal base with side lengths of 15 centimeters. When the hexagon is divided into six equal triangles, the height of each triangle is about 13 centimeters. If the aquarium is 50 centimeters tall, what is its volume?
\nAnswer:
\nIt is given that
\nAn aquarium has a regular hexagonal base with side lengths of 15 centimeters. When the hexagon is divided into six equal triangles, the height of each triangle is about 13 centimeters and the aquarium is 50 centimeters tall
\nNow,
\nFrom the given information,
\nWe can observe that the aquarium is in the form of a regular hexagonal prism
\nNow,
\nWe know that,
\nThe volume of any three-dimensional figure = Base area \u00d7 Height
\nWhere,
\nThe base area for this is the area of the hexagon
\nNow,
\nWe know that,
\nThe area of the hexagon (B) = 6 \u00d7 \\(\\frac{1}{2}\\) \u00d7 base \u00d7 Height
\n= 6 \u00d7 \\(\\frac{1}{2}\\) \u00d7 15 \u00d7 13
\n= 585 cm\u00b2
\nSo,
\nThe volume of the given aquarium (V) = 585 \u00d7 50
\n= 29,250 cm\u00b3
\nHence, from the above,
\nWe can conclude that the volume of the given aquarium is: 29,250 cm\u00b3<\/p>\n

Question 5.
\nA cheese box is shaped like a right triangular prism. The box is 6 inches long, 4 inches tall, and has a volume of 24 cubic inches. Can a cube of cheese that is 2.5 inches on each side fit inside the box? Explain.
\n\"Envision
\nAnswer:
\nIt is given that
\nA cheese box is shaped like a right triangular prism. The box is 6 inches long, 4 inches tall, and has a volume of 24 cubic inches.
\nNow,
\nWe know that,
\nThe volume of a right triangular prism (V) = Base area of a right triangle \u00d7 Height
\nNow,
\nWe know that,
\nArea of a triangle (A) = \\(\\frac{1}{2}\\) \u00d7 Base \u00d7 Height
\nSo,
\n24 = \\(\\frac{1}{2}\\) \u00d7 6 \u00d7 4 \u00d7 h
\n24 = 12 \u00d7 h
\nh = \\(\\frac{24}{12}\\)
\nh = 2 inches
\nNow,
\nFrom the above,
\nWe can observe that
\n2 inches < 2.5 inches
\nHence, from the above,
\nWe can conclude that a cube of cheese that is 2.5 inches on each side does not fit inside the box<\/p>\n

Question 6.
\nRay made a toolbox with the dimensions shown to store garden tools. What is the volume of the toolbox?
\n\"Envision
\nAnswer:
\nIt is given that
\nRay made a toolbox with the dimensions shown to store garden tools.
\nNow,
\nThe given figure is:
\n\"Envision
\nSo,
\nThe volume of the toolbox = (The volume of a cuboid) + (The volume of a trapezoidal prism)
\nNow,
\nWe know that,
\nThe volume of a cuboid (V) = Length \u00d7 Width \u00d7 Height
\nThe volume of a trapezoidal prism (V) = (Area of a trapezoid) \u00d7 (Height of the trapezoidal prism)
\nThe area of a trapezoid (A) = [(b 1<\/sub> + b 2 <\/sub>) h] \/ 2
\nSo,
\nThe volume of the toolbox = (16 \u00d7 3 \u00d7 5) + [(7 + 7) \u00d7 5] \u00f7 2 \u00d7 5
\n= (16 \u00d7 15) + \\(\\frac{70}{2}\\) \u00d7 5
\n= 240 + 175
\n= 415 in.\u00b3
\nHence, from the above,
\nWe can conclude that
\nThe volume of the toolbox is: 415 in.\u00b3<\/p>\n

Practice & Problem Solving<\/strong><\/p>\n

Leveled Practice In 7-8, find the volume of each prism.<\/p>\n

Question 7.
\n\"Envision
\nAnswer:
\nThe given figure is:
\n\"\"
\nNow,
\nFrom the given figure,
\nWe can observe that the given prism is a Right-triangular prism
\nSo,
\nThe volume of a right-triangular prism (V) = (Area of the right triangle) \u00d7 Height
\nSo,
\nThe volume of the given prism is given as:
\n\"\"
\nHence, from the above,
\nWe can conclude that
\nThe volume of the given prism is: 41.16 m\u00b3<\/p>\n

Question 8.
\n\"Envision
\nAnswer:
\nThe given figure is:
\n\"\"
\nNow,
\nFrom the given figure,
\nWe can observe that
\nThe given figure is: Hexagonal prism
\nSo,
\nThe volume of the given prism (V) = (The area of a hexagon) \u00d7 Height
\nNow,
\nWe know that,
\nThe area of a hexagon (A) = \\(\\frac{1}{2}\\) \u00d7 6 \u00d7 (Base of the triangle) \u00d7 (Height of the triangle)
\nSo,
\n\"\"
\nHence, from the above,
\nWe can conclude that
\nThe volume of the given prism is: 1,793.4 cm\u00b3<\/p>\n

Question 9.
\nA tunnel for an amusement park ride has the shape of a regular hexagonal prism with dimensions shown. The prism has a volume of 3,572.1 cubic meters. Can two 8-meter cars connected by a 3-meter connector pass through the tunnel at the same time? Explain.
\n\"Envision
\nAnswer:
\nIt is given that
\nA tunnel for an amusement park ride has the shape of a regular hexagonal prism with dimensions shown. The prism has a volume of 3,572.1 cubic meters.
\nNow,
\nThe given figure is:
\n\"Envision
\nNow,
\nFrom the given information,
\nWe can say that
\nThe volume of a regular hexagonal prism (V) = (Area of a hexagon) \u00d7 (Height of a prism)
\nNow,
\nWe know that,
\nThe area of a hexagon (A) = \\(\\frac{1}{2}\\) \u00d7 6 \u00d7 (Base of the triangle) \u00d7 (Height of the triangle)
\nSo,
\n3,572.1 = 6 \u00d7 \\(\\frac{1}{2}\\) \u00d7 7 \u00d7 8.1 \u00d7 h
\n3,572.1 = 170.1 \u00d7 h
\nh = \\(\\frac{3,572.1}{170.1}\\)
\nh = 21 m
\nNow,
\nThe area of the tunnel = 8 \u00d7 3
\n= 24 m\u00b2
\nSo,
\n24 > 21
\nHence, from the above,
\nWe can conclude that the two 8-meter cars connected by a 3-meter connector can’t pass through the tunnel at the same time<\/p>\n

Question 10.
\nA volume of 185.5 cubic feet of concrete was used to make the section of a skateboard ramp shown. How long is the ramp?
\n\"Envision
\nAnswer:
\nIt is given that
\nA volume of 185.5 cubic feet of concrete was used to make the section of a skateboard ramp shown.
\nNow,
\nThe given figure is:
\n\"Envision
\nNow,
\nWe know that,
\nThe volume of a rectangular prism (V) = Length \u00d7 Width \u00d7 Height
\nNow,
\nLet the length of the skateboard ramp be x ft
\nSo,
\nThe volume of a skateboard ramp (V) = 4 \u00d7 7 \u00d7 x
\n185.5 = 28 \u00d7 x
\nx = \\(\\frac{185.5}{28}\\)
\nx = 6.62
\nx \u2248 7 ft
\nHence, from the above,
\nWe can conclude that the length of the ramp is about 7 ft<\/p>\n

Question 11.
\nMake Sense and Persevere A small cube has a volume of 64 cubic feet. A larger cube has sides that are three times as long as the small cube. How long are the sides of each cube?
\n\"Envision
\nAnswer:
\nIt is given that
\nA small cube has a volume of 64 cubic feet. A larger cube has sides that are three times as long as the small cube
\nNow,
\nWe know that,
\nThe volume of a cube (V) = Side\u00b3
\nSo,
\nFor a small cube,
\nSide\u00b3 = 64
\nSide = \\(\\sqrt[3]{64}\\)
\nSide = 4 feet
\nNow,
\nAccording to the given information,
\nThe side of a larger cube = 3 \u00d7 (The side of a small cube)
\n= 3 \u00d7 4
\n= 12 feet
\nHence, from the above,
\nWe can conclude that
\nThe side length of a small cube is: 4 feet
\nThe side length of a larger cube is: 12 feet<\/p>\n

Question 12.
\nWhat is the volume of the regular hexagonal prism, to the nearest cubic centimeter?
\n\"Envision
\nAnswer:
\nThe given figure is:
\n\"Envision
\nNow,
\nWe know that,
\nThe volume of a regular hexagonal prism (V) = (Area of a hexagon) \u00d7 (Height of the prism)
\nNow,
\nWe know that,
\nArea of a hexagon (A) = \\(\\frac{1}{2}\\) \u00d7 6 \u00d7 (Base of the triangle) \u00d7 (Height of the triangle)
\nSo,
\nThe volume of the given prism (V) = 6 \u00d7 \\(\\frac{1}{2}\\) \u00d7 9 \u00d7 10.4 \u00d7 24
\n= 6,739.2
\n\u2248\u00a06,740 cm\u00b3
\nHence, from the above,
\nWe can conclude that the volume of the given regular hexagonal prism is: 6,740 cm\u00b3<\/p>\n

Question 13.
\nA mailbox has the dimensions shown What is the volume of the mailbox?
\n\"Envision
\nAnswer:
\nThe given figure is:
\n\"Envision
\nNow,
\nFrom the given mailbox,
\nWe can observe that
\nThe mailbox is the combination of a right triangular prism and a regular prism
\nNow,
\nWe know that,
\nThe volume of a right triangular prism (V) = (Area of a right triangle) \u00d7 (The height of a prism)
\nThe volume of a regular prism (V) = Length \u00d7 Width \u00d7 Height
\nNow,
\nWe knwo that,
\nThe area of a right triangle (A) = \\(\\frac{1}{2}\\) \u00d7 (Base of a triangle) \u00d7 (Height of a triangle)
\nSo,
\nThe volume of the given mailbox (V) = (\\(\\frac{1}{2}\\) \u00d7 2 \u00d7 8 \u00d7\u00a08) + (8 \u00d7 8 \u00d7 12)
\n= 64 + 768
\n= 832 in.\u00b3
\nHence, from the above,
\nWe can conclude that the volume of the given mailbox is: 832 in.\u00b3<\/p>\n

Question 14.
\nUse Structure A glass bead has the shape of a prism with a rectangular prism removed. What is the volume of the glass that forms the bead?
\n\"Envision
\nAnswer:
\nIt is given that
\nA glass bead has the shape of a prism with a rectangular prism removed.
\nNow,
\nThe given figure is:
\n\"Envision
\nNow,
\nThe volume of the glass that forms the bead (V) = (The volume of the glass bead that has the shape of a prism) + (The volume of a rectangular pris that is removed)
\n= (2 \u00d7 2 \u00d7 5) + (3 \u00d7 3 \u00d7 5)
\n= 20 + 45
\n= 65 cm\u00b3
\nHence, from the above,
\nWe can conclude that
\nThe volume of the glass that forms the bead is: 65 cm\u00b3<\/p>\n

Question 15.
\nHigher Order Thinking A cake has two layers. Each layer is a regular hexagonal prism. A slice removes one face of each prism, as shown.
\n\"Envision
\na. What is the volume of the slice?
\nAnswer:<\/p>\n

b. What is the volume of the remaining cake?
\nAnswer:<\/p>\n

Assessment Practice<\/strong>
\nQuestion 16.
\nThe area of the top of the box shown is 60 square centimeters. What is the volume, in cubic centimeters, of the box?
\n\"Envision
\nAnswer:
\nIt is given that
\nThe area of the top of the box shown is 60 square centimeters
\nNow,
\nThe given figure is:
\n\"Envision
\nSo,
\nThe volume of the given box (V) = 60 \u00d7 2 \u00d7 5
\n= 600 cm\u00b3
\nHence, from the above,
\nWe can conclude that
\nThe volume of the given box is: 600 cm\u00b3<\/p>\n

Question 17.
\nWhich of the following freezers is the best buy in terms of dollars per cubic foot?
\nA. Freezer A has an interior of 1 foot by 1 foot by 5 feet and sells for $499.99.
\nB. Freezer B has two sections of 3 cubic feet each and sells for $629.99.
\nC. Freezer C has interior dimensions of 1.5 feet by 1.5 feet by 4 feet and sells for $849.99.
\nD. Freezer D has 3 sections of 1.5 cubic feet each and sells for $444.99.
\nAnswer:
\nWe now that,
\nTo know which Freezer is best,
\nWe have to find the volume of the given Freezer and see whether for the high volume of Freezer, the cost of the Freezer is low or not
\nNow,
\nFor Freezer A,
\nVolume (V) = 1 \u00d7 1 \u00d7 5 = 5 ft\u00b3
\nThe ratio in terms of dollars per cubic foot = \\(\\frac{499.99}{5}\\)
\n= 100
\nFor Freezer B,
\nVolume (V) = 3 +\u00a03 \u00a0= 6 ft\u00b3
\nThe ratio in terms of dollars per cubic foot = \\(\\frac{629.99}{6}\\)
\n= 105
\nFor freezer C,
\nVolume (V) = 1.5 \u00d71.5 \u00d7 4
\n= 2.25\u00d74
\n= 9.00
\n= 9 ft\u00b3
\nThe ratio in terms of dollars per cubic foot = \\(\\frac{849.99}{9}\\)
\n= 94
\nFor Freezer D,
\nVolume (V) = 1.5 + 1.5 + 1.5 = 4.5 ft\u00b3
\nThe ratio in terms of dollars per cubic foot = \\(\\frac{444.99}{4.5}\\)
\n= 99
\nHence, from the above,
\nWe can conclude that
\nThe freezers that is the best buy in terms of dollars per cubic foot are:
\n\"\"<\/p>\n

Topic 8 Review<\/h3>\n

Topic Essential Question<\/strong>
\nHow can geometry be used to solve problems?
\nAnswer:
\nYou can use this field of mathematics to help you solve problems by drawing\u00a0your problem\u00a0and separating it out into geometric shapes. To do this, you divide your shape into smaller, common geometric shapes such as squares, rectangles, triangles, and circles and use the appropriate formulas for each<\/p>\n

Vocabulary Review<\/strong>
\nComplete each definition, and then provide an example of each vocabulary word.
\n\"Envision
\n\"Envision
\nAnswer:
\n\"\"<\/p>\n

Use Vocabulary in Writing<\/strong>
\nShawna drew this picture of three intersecting lines. Use vocabulary terms to explain how she could determine the value of x.
\n\"Envision
\nAnswer:
\nShawna drew this picture of three intersecting lines
\nNow,
\nThe given figure is:
\n\"Envision
\nNow,
\nFrom the given figure,
\nWe can observe that
\n47\u00b0 and x\u00b0 are the vertical angles
\nNow,
\nWe know that,
\nThe vertical angles have the same angle measures
\nSo,
\nx\u00b0 = 47\u00b0
\nHence, from the above,
\nWe can conclude that the value of x is: 47\u00b0<\/p>\n

Concepts and Skills Review<\/strong><\/p>\n

Lesson 8.1 Solve Problems involving Scale Drawings<\/strong><\/p>\n

Quick Review<\/strong>
\nIn a scale drawing, the scale is a ratio that relates each drawing length to the actual length it represents. To find unknown lengths, you can use the scale to write a proportion.<\/p>\n

Example
\nA blueprint of a room is drawn to a scale of 2 inches = 7 feet. The actual length of one wall is 56 feet. What is the length of this wall on the blueprint?
\nAnswer:
\nUse the scale to write a proportion.
\n\\(\\frac{2 \\text { in. }}{7 \\mathrm{ft}}=\\frac{x \\text { in. }}{56 \\mathrm{ft}}\\)
\n56 feet = 7 feet x 8, so multiply 2 inches by 8
\nx = 2 inches x 8 = 16 inches
\nThe length in the blueprint is 16 inches.<\/p>\n

Practice<\/strong>
\nUse the scale drawing to answer the questions.
\n\"Envision<\/p>\n

Question 1.
\nWhat is the actual area of the meeting room in square yards?
\nAnswer:
\nThe given figure is:
\n\"Envision
\nNow,
\nFrom the given figure,
\nWe can observe that
\nScale factor = \\(\\frac{3}{1}\\)
\nSo,
\nThe actual length of the meeting room = (Scale factor) \u00d7 (The length of the scale drawing)
\n= 3 \u00d7 4
\n= 12 yd
\nSo,
\nThe actual width of the meeting room = (Scale factor) \u00d7 (The width of the scale drawing)
\n= 3 \u00d7 1.5
\n= 4.5 yd
\nSo,
\nThe actual area of the meeting room = (The actual length of the meeting room) \u00d7 (The actual width of the meeting room)
\n= 12 \u00d7 4.5
\n= 54.0
\n= 54 yd\u00b2
\nHence, from the above,
\nWe can conclude that
\nThe actual area of the meeting room is: 54 yd\u00b2<\/p>\n

Question 2.
\nJohn decides to make a new scale drawing of the meeting room. He wants the length of the room in the new drawing to be 8 inches. What is the new scale for the drawing?
\nAnswer:
\nIt is given that
\nJohn decides to make a new scale drawing of the meeting room. He wants the length of the room in the new drawing to be 8 inches
\nNow,
\nFrom Question 1,
\nThe actual length of the meeting room = (Scale factor) \u00d7 (The length of the scale drawing)
\n= 3 \u00d7 4
\n= 12 yd
\nSo,
\n\\(\\frac{1}{8}\\) = \\(\\frac{12}{x}\\)
\nx = 96
\nSo,
\nThe new scale for the drawing is:
\n8 inches = 96 yd
\nDivide by 8 on both sides
\n1 inch = 12 yd
\nHence, from the above,
\nWe can conclude that
\nThe new scale for the drawing is:
\n1 inch = 12 yd<\/p>\n

Lesson 8.2 Draw Geometric Figures<\/strong><\/p>\n

Quick Review<\/strong>
\nYou can classify a quadrilateral as a trapezoid, a rectangle, a square, or a parallelogram based on its side lengths, side relationships, and angle measures.<\/p>\n

Example
\nDraw a quadrilateral with exactly two perpendicular sides and one angle measuring 120\u00b0. What is the name of the figure you drew?
\nAnswer:
\n\"Envision
\nThe figure has exactly one pair of parallel sides, so it is a trapezoid.<\/p>\n

Practice<\/strong>
\nQuestion 1.
\nDraw a quadrilateral with two pairs of parallel sides, with one side measuring 5 centimeters, one side measuring 3 centimeters, and one angle measuring 45\u00b0. What is the name of the figure you drew?
\nAnswer:
\nThe representation of the quadrilateral with two pairs of parallel sides, with one side measuring 5 centimeters, one side measuring 3 centimeters, and one angle measuring 45\u00b0 is:
\n\"\"
\nNow,
\nFrom the figure,
\nWe can observe that
\na. The side lengths of the parallel sides are the same
\nb. The angle measures are not 90\u00b0
\nHence, from the above,
\nWe can conclude that
\nThe quadrilateral with the given specifications is called: parallelogram<\/p>\n

Question 2.
\nWhat quadrilaterals can you draw that have two angles measuring 115\u00b0 and two angles measuring 65\u00b0?
\nAnswer:
\nIt is given that the two angles measuring 115\u00b0 and two angles measuring 65\u00b0
\nSo,
\nThe representation of the quadrilateral that has two angles measuring 115\u00b0 and two angles measuring 65\u00b0 is:
\n\"\"
\nHence, from the above,
\nWe can conclude that
\nThe quadrilateral that you can draw with the same opposite angle measures is: Parallelogram<\/p>\n

Lesson 8.3 Draw Triangles with Given Conditions<\/strong><\/p>\n

Quick Review<\/strong>
\nWhen you are given certain conditions for a triangle, it may be possible to draw one triangle, more than one triangle, or no triangle.<\/p>\n

Example
\nHow many triangles can be drawn with side lengths of 3 inches, 5 inches, and 6 inches?
\nAnswer:
\nNo matter how you position the sides, the triangle has the same shape and size. There is only one way to draw a triangle with these side lengths.
\n\"Envision<\/p>\n

Practice<\/strong>
\nQuestion 1.
\nCan more than one triangle be drawn with side lengths of 4 centimeters and 2 centimeters and an included angle of 50\u00b0? Explain.
\nAnswer:
\nIt is given that
\none triangle is drawn with side lengths of 4 centimeters and 2 centimeters and an included angle of 50\u00b0
\nSo,
\nThe representations of the triangles with the given specifications are:
\n\"\"
\nNow,
\nFrom the above triangles,
\nWe can observe that both the representations of the triangles are the same
\nHence, from the above,
\nWe can conclude that more than one triangle can not be drawn from the given specifications<\/p>\n

Question 2.
\nWhen given angle measures of 25\u00b0, 65\u00b0, and 90\u00b0, is it impossible to draw a triangle, possible to draw only one triangle, or possible to draw more than one triangle? Explain.
\nAnswer:
\nWe know that,
\nThere is more than one triangle possible for the below cases:
\na. Three different angle measures
\nb. Two sides and an included triangle
\nNow,
\nIn this question,
\nThree different angle measures are given
\nHence, from the above,
\nWe can conclude that\u00a0 it is impossible to draw more than one triangle<\/p>\n

Lesson 8-4 Solve Problems Using Angle Relationships<\/strong><\/p>\n

Quick Review<\/strong>
\nAngles that have a common vertex and a common side but no common interior points are adjacent angles. Supplementary angles are angles with a sum of 180\u00b0. Complementary angles are angles with a sum of 90\u00b0. When two lines intersect, the angles that have no side in common are called vertical angles. Vertical angles are equal.<\/p>\n

Example
\nList all pairs of vertical angles in this figure.
\nAnswer:
\n\"Envision
\nThere are two pairs of vertical angles:
\n\u2022 \u2220SZR and \u2220XZY
\n\u2022 \u2220SZX and \u2220RZY<\/p>\n

Practice<\/strong><\/p>\n

Use the figure from the example.
\n\"Envision
\nQuestion 1.
\nName a pair of complementary angles.
\nAnswer:
\nThe given figure is:
\n\"Envision
\nNow,
\nWe know that,
\nThe complementary angles are the angles that have the angle measure of 90\u00b0
\nHence, from the above,
\nWe can conclude that
\nThe pair of Complementary angles are: \u2220QZX and \u2220XZY<\/p>\n

Question 2.
\nThe measure of \u2220XZY is 55\u00b0. Which other angle has a measure of 55\u00b0? Explain.
\nAnswer:
\nIt is given that the angle measure of \u2220XZY is: 55\u00b0
\nNow,
\nThe given figure is:
\n\"Envision
\nNow,
\nFrom the given figure,
\nWe can observe that
\nThere are two pairs of vertical angles:
\na. \u2220SZR and \u2220XZY
\nb. \u2220SZX and \u2220RZY
\nNow,
\nWe know that,
\nThe vertical angles are the angles that have the same angle measures
\nHence, from the above,
\nWe can conclude that
\nThe other angles that have a measure of 55\u00b0 are: \u2220SZR, \u2220SZX, and \u2220RZY<\/p>\n

Question 3.
\nUse the information from Problem 2. Find the value of n.
\nAnswer:
\nThe given figure is:
\n\"Envision
\nNow,
\nFrom problem 2,
\nWe can observe that
\n\u2220SZR and \u2220RZY are the adjacent angles
\nSo,
\nAccording to the given information,
\n(3n + 5)\u00b0 + 55\u00b0 = 180\u00b0
\n3n\u00b0 = 180\u00b0 – 60\u00b0
\n3n\u00b0 = 120\u00b0
\nn = \\(\\frac{120\u00b0}{3}\\)
\nn = 40\u00b0
\nHence, from the above,
\nWe can conclude that the value of n is: 40\u00b0<\/p>\n

Lesson 8.5 Solve Problems Involving Circumference of a Circle<\/strong><\/p>\n

Quick Review<\/strong>
\nThe distance around a circle is called its circumference. The number a (pi) is the ratio of the circumference of any circle to its diameter. So when you know the diameter, d, of a circle, or its radius, r, you can determine its circumference, C, with the formula C = \u03c0d or C = 2\u03c0r.<\/p>\n

Example
\nWhat is the circumference of a circle with a radius of 6 meters? Use 3.14 for \u03c0.
\nAnswer:
\nC = 2\u03c0r
\nC = 2\u03c0(6)
\nC \u2248 2(3.14)(6)
\nC = 37.68
\nThe circumference is about 37.68 meters.<\/p>\n

Practice<\/strong><\/p>\n

Question 1.
\nThe length of the minute hand of a clock is 14 inches. What is the length of the path traced by the outer tip of the minute hand in one hour? Use \\(\\frac{22}{7}\\) for \u03c0.
\nAnswer:
\nIt is given that
\nThe length of the minute hand of a clock is 14 inches
\nSo,
\nThe length of the path traced by the outer tip of the minute hand in one hour = 2\u03c0 \u00d7 (The length of the minute hand of a clock)
\n= 2 \u00d7 \\(\\frac{22}{7}\\) \u00d7 14
\n= 88 inches
\nHence, from the above,
\nWe can conclude that
\nThe length of the path traced by the outer tip of the minute hand in one hour is: 88 inches<\/p>\n

Question 2.
\nThe circumference of a bicycle tire is 126.5 centimeters. What is the diameter of the tire? Use 3.14 for a. Round to the nearest tenth as needed.
\nAnswer:
\nIt is given that
\nThe circumference of a bicycle tire is 126.5 centimeters
\nNow,
\nWe know that,
\nThe circumference of a bicycle tire (C) = \u03c0d
\nWhere,
\nd is the diameter of the bicycle tire
\nSo,
\nAccording to the given information,
\n126.5 = 3.14 \u00d7 d
\nd = \\(\\frac{126.5}{3.14}\\)
\nd = 40.28
\nd \u2248 40.3 centimeters
\nHence, from the above,
\nWe can conclude that
\nThe diameter of the tire is: 40.3 centimeters<\/p>\n

Lesson 8.6 Solve Problems involving Area of a Circle<\/strong><\/p>\n

Quick Review<\/strong>
\nThe area, A, of a circle can be found using the formula A = \u03c0r2<\/sup>, where r is the radius. You can use 3.14 or \\(\\frac{22}{7}\\) as an approximation for \u03c0.<\/p>\n

Example
\nThe diameter of the logo at the center of a basketball court is 10 feet. What is the area of the logo? Use 3.14 for \u03c0.
\nAnswer:
\nThe radius of a circle is half the diameter. So the radius of the logo is half of 10 feet, or 5 feet. Substitute the radius into the circle area formula.
\nA = \u03c0r2<\/sup>
\nA = \u03c0(5)2<\/sup>
\nA \u2248 3.14(25)
\nA = 78.5
\nThe area of the logo is about 78.5 square feet.<\/p>\n

Practice<\/strong><\/p>\n

Question 1.
\nJessie wants to paint the top of the table shown. What is the approximate area that she will paint? Use 3.14 for it. Round to the nearest whole number of inches.
\n\"Envision
\nAnswer:
\nIt is given that
\nJessie wants to paint the top of the table shown.
\nNow,
\nThe given figure is:
\n\"Envision
\nNow,
\nWe know that,
\nRadius (r) = \\(\\frac{Diameter}{2}\\)
\nSo,
\nr = \\(\\frac{78}{2}\\)
\nr = 39 in
\nNow,
\nWe know that,
\nThe area of a circle (A) = \u03c0r\u00b2
\nSo,
\nThe area of the top of table (A) = 3.14 \u00d7 39\u00b2
\n= 4,775.94 in\u00b2
\n\u2248 4,776 in\u00b2
\nHence, from the above,
\nWe can conclude that the approximate area that she will paint is: 4,776 in.\u00b2<\/p>\n

Question 2.
\nWhat is the diameter of a circle with an area of 113.04 square centimeters? Use 3.14 for \u03c0.
\nAnswer:
\nIt is given that
\nThe area of a circle is: 113.04 square centimeters
\nNow,
\nWe know that,
\nThe area of a circle (A) = \u03c0r\u00b2
\nSo,
\n113.04 = 3.14 \u00d7 r\u00b2
\nr\u00b2 = \\(\\frac{113.04}{3.14}\\)
\nr\u00b2 = 36
\nr = 6 cm
\nNow,
\nWe know that,
\nDiameter (d) = 2 \u00d7 Radius (r)
\nSo,
\nd = 2 \u00d7 6
\nd = 12 cm
\nHence, from the above,
\nWe can conclude that
\nThe diameter of the given circle is: 12 cm<\/p>\n

Question 3.
\nThe distance around a circular park is 88 yards. What is the area of the park? Use \\(\\frac{22}{7}\\) for \u03c0.
\nAnswer:
\nIt is given that
\nThe distance around a circular park is 88 yards
\nNow,
\nWe know that,
\nCircumference (C) = 2\u03c0r
\nSo,
\n88 = 2 \u00d7 \\(\\frac{22}{7}\\) \u00d7 r
\nr = \\(\\frac{7 \u00d7 88}{2 \u00d7 22}\\)
\nr = 14 yards
\nNow,
\nWe know that,
\nArea of a circle (A) = \u03c0r\u00b2
\nSo,
\nThe area of the circular park (A) = \\(\\frac{22}{7}\\) \u00d7 14\u00b2
\n= 616 yards\u00b2
\nHence, from the above,
\nWe can conclude that the area of the given circular park is: 616 yards\u00b2<\/p>\n

Lesson 8.7 Describe Cross Sections<\/strong><\/p>\n

Quick Review<\/strong>
\nA cross section is the two-dimensional shape exposed when a three-dimensional figure is sliced. Recognizing the shape of a cross section can help in solving some problems.<\/p>\n

Example
\nMuffins are packed in two layers in a box with a piece of cardboard placed between. What shape is the cardboard and what are its dimensions?
\nAnswer:
\n\"Envision
\nThe cardboard lies on a cross section shaped like a rectangle that is 15 inches long and 10 inches wide.<\/p>\n

Practice<\/strong><\/p>\n

Question 1.
\nThe figure shows a vertical cross-section of a right rectangular pyramid. What shape is the cross-section and what is its area?
\n\"Envision
\nAnswer:
\nIt is given that
\nThe figure shows a vertical cross-section of a right rectangular pyramid
\nNow,
\nThe given figure is:
\n\"Envision
\nNow,
\nFrom the above figure,
\nWe can observe that
\nThe vertical cross-section of the given pyramid is: Right triangle
\nNow,
\nWe know that,
\nThe area of a right triangle (A) = \\(\\frac{1}{2}\\) \u00d7 (Base of the triangle) \u00d7 (Height of the triangle)
\nSo,
\nThe area of the vertical cross-section (A) = \\(\\frac{1}{2}\\) \u00d7 7 \u00d7 5
\n= 17.5 cm\u00b2
\nHence, from the above,
\nWe can conclude that
\nThe shape of the cross-section is: Right triangle
\nThe area of the cross-section is: 17.5 cm\u00b2<\/p>\n

Question 2.
\nZach wants to slice the pyramid along a horizontal plane that intersects the pyramid above its base. Describe the cross section that would be formed.
\nAnswer:
\nIt is given that
\nZach wants to slice the pyramid along a horizontal plane that intersects the pyramid above its base
\nNow,
\nThe given figure is:
\n\"Envision
\nHence, from the above,
\nWe can conclude that
\nThe shape of the horizontal plane when the pyramid is intersected is: Rectangle<\/p>\n

Lesson 8.8 Solve Problems involving Surface Area<\/strong><\/p>\n

Quick Review<\/strong>
\nA composite figure is the combination of two or more geometric shapes. The surface area of a two- and a three-dimensional composite figure will be the sum of the areas of all the shapes, or faces.<\/p>\n

Example
\nThe figure shows the plan for a kitchen countertop. What is the area of the countertop?
\n\"Envision
\nAnswer:
\n(3 \u2022 8) + (3 \u2022 2) + \\(\\frac{1}{2}\\)(3 \u2022 3) = 24 + 6 + 4.5 = 34.5
\nThe area of the countertop is 34.5 ft2<\/sup>.<\/p>\n

Practice<\/strong><\/p>\n

Question 1.
\nKara wants to paint the four outside walls of her dog’s house. She will not paint the roof or the door on the front of the house. What is the area of the surface that Kara needs to paint?
\n\"Envision
\nAnswer:
\nIt is given that
\nKara wants to paint the four outside walls of her dog’s house. She will not paint the roof or the door on the front of the house.
\nNow,
\nThe given figure is:
\n\"Envision
\nNow,
\nThe area of the dog’s house that Kara needed to paint (A) = (The total area of the dog’s house) – (The area of the front of the house)
\n= ([\\(\\frac{1}{2}\\) \u00d7 4 \u00d7 4] + [7 \u00d7 3]) – (\\(\\frac{1}{2}\\) \u00d7 4 \u00d7 4)
\n= 29 – 8
\n= 21 ft\u00b2
\nHence, from the above,
\nWe can conclude that
\nThe area of the surface that Kara needs to paint is: 21 ft\u00b2<\/p>\n

Lesson 8.9 Solve Problems involving Volume<\/strong><\/p>\n

Quick Review<\/strong>
\nYou can find the volume, V, of a prism using the formula V = Bh. In this formula, B represents the area of the base of the prism and h represents the height of the prism. Volume is measured in cubic units.<\/p>\n

If the volume of a prism is known, you may be able to use this formula to find an unknown dimension of the prism. You also can use this formula to solve problems involving volumes of composite figures that are made up of two or more prisms.<\/p>\n

Example
\nRhonda received a package in a box shaped like a rectangular prism. What is the volume of the box?
\n\"Envision
\nAnswer:
\nFind the area of the rectangular base.
\nA = 9(18) = 162 in.2<\/sup>
\nFind the volume of the prism.
\nV = Bh
\nV= 162(9)
\nV= 1,458 in.3<\/sup><\/p>\n

Practice<\/strong><\/p>\n

Question 1.
\nHolly has a gift box that is shaped like a regular hexagonal prism. What is the volume of the box?
\n\"Envision
\nAnswer:
\nIt is given that
\nHolly has a gift box that is shaped like a regular hexagonal prism
\nNow,
\nThe given figure is:
\n\"Envision
\nNow,
\nWe know that,
\nThe volume of a regular hexagonal prism (V) = (Area of a hexagon) \u00d7 (Height of the regular hexagonal prism)
\nThe area of a hexagon (A) = 6 \u00d7 \\(\\frac{1}{2}\\) \u00d7 (Base of a triangle}{Height of a triangle}
\nSo,
\nV = 6 \u00d7 \\(\\frac{1}{2}\\) \u00d7 7.8 \u00d7 12 \u00d7 9
\n= 2,527.2 in.\u00b3
\nHence, from the above,
\nWe can conclude that
\nThe volume of the given regular hexagonal prism is: 2,527.2 in.\u00b3<\/p>\n

Question 2.
\nA designer is planning a trail mix box that is shaped like a rectangular prism. The front of the box must have the width and height shown. The volume of the box must be 162 cubic inches. What must be the depth, d, of the box?
\n\"Envision
\nAnswer:
\nIt is given that
\nA designer is planning a trail mix box that is shaped like a rectangular prism. The front of the box must have the width and height shown. The volume of the box must be 162 cubic inches.
\nNow,
\nThe given figure is:
\n\"Envision
\nNow,
\nWe know that,
\nThe volume of a rectangular prism (V) = Length \u00d7 Width \u00d7 Height
\nSo,
\n162 = 9 \u00d7 7.5 \u00d7 d
\n162 = 67.5 \u00d7 d
\nd = \\(\\frac{162}{67.5}\\)
\nd = 2.4 in.
\nHence, from the above,
\nWe can conclude that
\nThe depth of the box is 2.4 in.<\/p>\n

Question 3.
\nA building that is used for storage has the dimensions shown. What is the volume of the building?
\n\"Envision
\nAnswer:
\nIt is given that
\nA building that is used for storage has the dimensions shown
\nNow,
\nThe given figure is:
\n\"Envision
\nSo,
\nThe volume of the building (V) = (8 \u00d7 28 \u00d714) + (12 \u00d7 16 \u00d7 14)
\n= 3,136 + 2,688
\n= 5,824 ft\u00b3
\nHence, from the above,
\nWe can conclude that
\nThe volume of the building is: 5,824 ft\u00b3<\/p>\n

Topic 8 Fluency Practice<\/h3>\n

Pathfinder
\nShade a path from START to FINISH. Follow the answers to the problems so that each answer is greater than the one before. You can only move up, down, right, or left.
\nI can… use the percent equation to solve problems.<\/p>\n

START<\/strong>
\n\"Envision<\/p>\n","protected":false},"excerpt":{"rendered":"

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