Looking Ahead to Chapter 7

Transcription

1 Looking Ahead to Chapter Focus In Chapter, you will learn how to identify and find unknown measures in similar polygons and solids, prove that two triangles are similar, and use indirect measurement to solve problems. Chapter Warmup Answer these questions to help you review skills that you will need in Chapter. Write each fraction in simplest form Multiply. Write your answer in simplest form Solve each proportion. x x 4 50 x 200 Read the problem scenario below. You look up into the sky and see an airplane flying toward the airport. The airplane is 22,16 feet directly above you. You are 55 miles from the airport. 10. How far, in miles, are you from the airplane? 11. About how far, in miles, is the airplane from the airport? Key Terms ratio p. 284 proportion p. 28 means p. 28 extremes p. 28 similar p. 290 congruent p. 290 scale model p. 295 scale p. 295 paragraph proof p. 304 indirect measurement p. 305 similar solids p. 312 scale factor p Chapter Similarity

2 CHAPTER Similarity The first plastic containers for storing food were introduced in These containers grew in popularity because they were more durable than glassware and kept food fresh longer. In Lesson.5, you will compare the surface area and volume of similar plastic containers..1 Ace Reporter Ratio and Proportion p Framing a Picture Similar and Congruent Polygons p Using an Art Projector Proving Triangles Similar: AA, SSS, and SAS p Modeling a Park Indirect Measurement p Making Plastic Containers Similar Solids p. 311 Chapter Similarity 281

3 282 Chapter Similarity

4 .1 Objectives In this lesson, you will: Write and simplify ratios. Compare ratios. Write and solve proportions. Use survey results to make predictions. Key Terms ratio proportion means extremes Ace Reporter Ratio and Proportion SCENARIO You are a reporter for your school s newspaper. You are writing an article about the order of classes during the school day and you are interviewing students to see what they think. Problem 1 Survey Says From the investigating you have done so far, it seems that the students have a strong opinion on when the physical education class should occur. You have surveyed many students and recorded the results in the table below. When Do You Think Gym Classes Should Be Held? Beginning of Day End of Day Any Time A. How many students did you survey? Show all your work and use a complete sentence in your answer. B. What can you conclude from your survey results? Explain your reasoning. Use complete sentences in your answer. Investigate Problem 1 1. In your article you want to compare the results in your survey. One way you could compare the results is by writing the statement, Eight out of 24 students prefer to have gym class at the beginning of the day. Which two numbers from the survey results are being compared? Use a complete sentence in your answer. Lesson.1 Ratio and Proportion 283

5 Investigate Problem 1 Complete the following statements that compare the numbers in the survey. out of 24 students prefer to have gym class at the end of the day. Two out of 24 students. 2. Just the Math: Ratio You can mathematically compare the results in the table by using ratios. A ratio is a comparison of two numbers that uses division. You can write a ratio as a fraction or by using a colon. For instance, you can write Eight out of 24 students prefer to have gym class at the beginning of the day in two ways. 8 students As a fraction: 24 students Using a colon: 8 students : 24 students When you use a colon, you read the colon as the word to. So, the statement 8 students : 24 students is read as 8 students to 24 students. Write each of the other statements from Question 1 as a ratio. Write each ratio as a fraction. If possible, simplify your fractions. 3. Suppose that you have only surveyed students in your own grade. A friend of yours offers to help you out and surveys students from another grade in your school. Your friend s results are shown in the table below. When Do You Think Gym Classes Should Be Held? Beginning of Day End of Day Any Time How many students did your friend survey? Write a ratio that compares the number of students that prefer gym class at the beginning of the day to the number of students surveyed. Write your answer as a fraction in simplest form. 284 Chapter Similarity

6 Investigate Problem 1 Write a ratio that compares the number of students that prefer gym class at the end of the day to the number of students surveyed. Write your answer as a fraction in simplest form. Write a ratio that compares the number of students that do not care when gym class is held to the number of students surveyed. Write your answer as a fraction in simplest form. 4. Do a larger portion of the students in your survey or your friend s survey prefer to have gym class at the beginning of the day? Show all your work and explain your reasoning. Use complete sentences in your answer. Do a larger portion of the students in your survey or your friend s survey prefer to have gym class at the end of the day? Show all your work and explain your reasoning. Use complete sentences in your answer. Do a larger portion of the students in your survey or your friend s survey have no preference for when gym class is held? Show all your work and explain your reasoning. Use complete sentences in your answer. 5. When are two different ratios equivalent? Use complete sentences to explain your reasoning. Lesson.1 Ratio and Proportion 285

7 Investigate Problem 1 6. Complete the table below to show the results of your survey and your friend s survey together. Then write two equivalent ratios for each statement. Write your ratios as fractions. When Do You Think Gym Classes Should Be Held? Beginning of Day End of Day Any Time Students who prefer gym class at end of day : Students who prefer gym class at beginning of day Students who have no preference : Students who prefer gym class at end of day Problem 2 Making Predictions A. Use the combined results of the surveys in Problem 1 to write the following ratios. Write each ratio as a fraction in simplest form. Students who prefer gym class at beginning of day : All students surveyed Students who prefer gym class at end of day : All students surveyed Students with no preference : All students surveyed 286 Chapter Similarity

8 Problem 2 Making Predictions B. Suppose that you want to interview students from the other grades in your school. Would you expect that the results you would get from surveying the other grades would be very different from the results you already have? Why or why not? Use complete sentences in your answer. C. Suppose that you interviewed 30 students in a different grade. How many students would you expect to respond that they prefer to have gym class at the end of the day? Explain your reasoning. Use complete sentences in your answer. Investigate Problem 2 1. Just the Math: Proportion When two ratios that compare the same quantities are equal, you can write them as a proportion. A proportion is an equation that states that two ratios are equivalent, or equal. We write a proportion by placing an equals sign between two equivalent ratios or by using a double colon in place of the equals sign. For instance, you could have used a proportion to answer part (C): 8 students? students. 15 students 30 students What is the value of the unknown quantity in the proportion above? Use complete sentences to explain how you found your answer. When you found the unknown quantity, you were solving the proportion. 2. Another way to solve a proportion is by using the proportion s means and extremes. extremes a b c d means Lesson.1 Ratio and Proportion 28

9 Investigate Problem 2 What are the means of the solved proportion in Question 1? What are the extremes of the solved proportion in Question 1? Use complete sentences in your answer. Find the product of the means and the product of the extremes from Question 1. What do you notice? Show all your work and use a complete sentence in your answer. Use the results in Question 2 to complete the steps to solve the following proportion. Show all your work x 18 x Set product of extremes equal to product of means. Divide each side by 4. Simplify. Use complete sentences to explain how to solve a proportion by using the proportion s means and extremes. 3. Suppose that there are 480 students in your school. Use the combined survey results from Problem 1 to predict how many students in your school would prefer to have gym class at the beginning of the day, how many students would prefer to have gym class at the end of the day, and how many students have no preference. Show all your work and use complete sentences in your answer. 288 Chapter Similarity

10 .2 Objectives In this lesson, you will: Identify similar and congruent polygons. Identify corresponding angles and corresponding sides in similar and congruent polygons. Find unknown measures in similar and congruent polygons. Find unknown measures in a scale model. Key Terms similar congruent scale model scale Framing a Picture Similar and Congruent Polygons SCENARIO When you frame a picture, it is not unusual to put a mat inside the frame. A mat is a piece of paperboard that is used to provide a transition between a picture and the picture frame. Problem 1 The Perfect Picture You are creating your own collage of pictures. You have bought a large frame and will cut out rectangular holes in the mat as shown. 4 in. 6 in. in. D 5 in. C frame mat picture 3 in. A 3 in. B 2 in. 2 in. A. What are the interior angle measures of each mat opening? Use a complete sentence in your answer. Take Note In this lesson, the length refers to the longer side of the rectangle. B. Write a ratio that compares the length of rectangle A to the length of rectangle B. Then write a ratio that compares the width of rectangle A to the width of rectangle B. Write your answers as fractions in simplest form. What do you notice? Use a complete sentence in your answer. Lesson.2 Similar and Congruent Polygons 289

11 Problem 1 The Perfect Picture C. Write a ratio that compares the length of rectangle A to the length of rectangle D. Then write a ratio that compares the width of rectangle A to the width of rectangle D. Write your answers as fractions in simplest form. What do you notice? Use a complete sentence in your answer. D. Write a ratio that compares the length of rectangle A to the length of rectangle C. Then write a ratio that compares the width of rectangle A to the width of rectangle C. What do you notice? Use a complete sentence in your answer. Investigate Problem 1 1. Two polygons are similar when the corresponding angles are congruent and the ratios of the measures of the corresponding sides are equal. Which rectangles from Problem 1 are similar? Explain your reasoning. Use complete sentences in your answer. Take Note In a rectangle, one pair of corresponding sides are the lengths and the other pair of corresponding sides are the widths. 2. Two polygons are congruent when the corresponding angles are congruent and the corresponding sides are congruent. Which rectangles from Problem 1 are congruent? Explain your reasoning. Use complete sentences in your answer. 290 Chapter Similarity

12 Investigate Problem 1 3. What do you notice about the ratios of the corresponding sides of congruent figures? Use a complete sentence in your answer. 4. Are all similar figures also congruent figures? If so, explain your reasoning. If not, give an example that shows a pair of similar figures that are not congruent. Use complete sentences in your answer. 5. Are all congruent figures also similar figures? If so, explain your reasoning. If not, give an example that shows a pair of congruent figures that are not similar. Use complete sentences in your answer. 6. The triangles shown below are congruent. B E A C D F You can write ABC DEF. Whenever you write a congruence statement like this, the letters that name the vertices should be written in corresponding order. For instance, A D, so A and D are in the same position. Name the pairs of corresponding angles and corresponding sides. The measure of A is 40º; the measure of E is 88º; the length of AB is 2.3 centimeters; and the length of DF is 2.8 centimeters. Label this information on the figures above. What is the length of DE? Explain how you found your answer. Use a complete sentence in your answer. What is the measure of B? Explain how you found your answer. Use a complete sentence in your answer. Lesson.2 Similar and Congruent Polygons 291

13 Investigate Problem 1. The two triangles below are similar. You can write UVW ~ XYZ, where the symbol ~ means is similar to. V Y U W X Z Again, the order in which you write the vertices in a similarity statement indicates the corresponding angles and the corresponding sides. List the corresponding angles and the corresponding sides. Write a ratio that compares a side length of UVW to a corresponding side length of XYZ. Then write a ratio that compares a side length of XYZ to a corresponding side length of UVW. Are the two ratios equal? Why or why not? Use complete sentences in your answer. Because the triangles are similar, we can write a proportion that relates the ratios of the lengths of the sides. One possible UV VW proportion is XY YZ. XZ YZ Another possible proportion is UW VW. When you write a proportion relating the corresponding side lengths of two similar polygons, what must be true about both of the ratios? Use complete sentences in your answer. 292 Chapter Similarity

14 Investigate Problem 1 8. In the figure below, GHI ~ KLM. L H G I K M Complete the following proportions that relate the ratios of the lengths of the sides of the triangle. GH KL HI KM GI LM LM GI KM Suppose that GH 3 feet, KL 9 feet, and HI 5 feet. Write a proportion that you can use to find LM. Then solve the proportion. Show all your work and use a complete sentence in your answer. Suppose that you also know that KM 12 feet. Find GI. Show all your work and use a complete sentence in your answer. Lesson.2 Similar and Congruent Polygons 293

15 Investigate Problem 1 Find the ratio of the height of GHI to the height of KLM. Find the ratio of the length of the base of GHI to the length of the base of KLM. Compare the ratios of the lengths and heights. Use a complete sentence in your answer. Find the areas of the triangles. Then find the ratio of the area of GHI to the area of KLM. Write your ratio as a fraction in simplest form. How is the ratio of the areas related to the ratio of the heights? How is the ratio of the areas related to the ratio of the lengths? Why do you think this is so? Explain your reasoning. Use complete sentences in your answer. 294 Chapter Similarity

16 Investigate Problem 1 9. A scale model (or model) of an object is similar to the actual object but is either larger or smaller. The ratio of a dimension of the actual object to a corresponding dimension in the model is called the scale of the model. A wall mural is being created from a picture that is 6 inches long and 4 inches wide. The wall mural should be 48 inches long. Complete the statement below to find the scale of the model. Write your answer as a fraction in simplest form. length of picture length of mural Now use the scale to complete the proportion that you can use to find the width of the mural. width of picture width of mural Find the width of the mural. Show all your work and use a complete sentence in your answer. 10. A scale model of a framed picture is being created for a dollhouse. The actual rectangular picture is 4 inches wide and 8 inches long. The scale of the model is 4 : 1. Find the length and width of the dollhouse picture. Show all your work and use a complete sentence in your answer. Lesson.2 Similar and Congruent Polygons 295

17 296 Chapter Similarity

18 .3 Objectives In this lesson, you will: Use given information to show that two triangles are similar. Complete a paragraph proof. Using an Art Projector Proving Triangles Similar: AA, SSS, and SAS SCENARIO An art projector is a piece of equipment that artists use to create exact copies of artwork, to enlarge artwork, or to reduce artwork. A basic art projector uses a light bulb and a lens within a box. The light rays from the art being copied are collected onto a lens at a single point. Then the lens projects the image of the art onto a screen as shown below. Art Projector Image Key Term paragraph proof Light Lens Screen If the projector is set up properly, the triangles above will be similar polygons. You can show that these triangles are similar without measuring all of the side lengths and all of the interior angles. Problem 1 Angles, Angles, Angles A. Suppose that two triangles are similar. What do you know about the two triangles? Use a complete sentence in your answer. B. Consider the two triangles below. Without using a ruler or protractor, can you determine whether the triangles are similar? Why or why not? Use complete sentences in your answer. Lesson.3 Proving Triangles Similar: AA, SSS, and SAS 29

19 Problem 1 Angles, Angles, Angles C. Use a protractor to create two different triangles that each have two interior angles that measure 35º and 45º. Label the vertices of your triangles. D. What do you know about the third interior angle in each of the triangles in part (C)? Explain how you found your answer. Use a complete sentence in your answer. E. Measure the side lengths of each triangle in part (C) to the nearest millimeter and record these lengths below. Also record the interior angle measures below. F. Are the triangles similar? Explain your reasoning and use a complete sentence in your answer. G. When the corresponding angles of two triangles are congruent, what can you conclude about the two triangles? Use a complete sentence in your answer. H. If you know that two pairs of corresponding angles are congruent, can you conclude that the triangles are similar? Why or why not? Use complete sentences in your answer. 298 Chapter Similarity

20 Take Note In the figure at the right, the double arcs show that C and F are congruent, but that these angles are not congruent to A and D. Investigate Problem 1 1. The result of Problem 1 is called the Angle-Angle Similarity Theorem. Angle-Angle Similarity Theorem If two angles of one triangle are congruent to two angles of another triangle, then the triangles are similar. A B C D E If m A m D and m C m F, then ABC ~ DEF. F Why do you think that it is enough to know that the corresponding angles of two triangles are congruent in order to say that the triangles are similar? In other words, why do you not need any information about the side lengths? Use complete sentences in your answer. 2. The triangles shown are isosceles triangles. Do you have enough information to show that the triangles are similar? Explain your reasoning. Use a complete sentence in your answer. M Q L N P R Lesson.3 Proving Triangles Similar: AA, SSS, and SAS 299

21 Take Note Investigate Problem 1 3. The triangles shown are isosceles triangles. Do you have enough information to show that the triangles are similar? Explain your reasoning. Use complete sentences in your answer. T W In the figure, the single hash marks indicate that ST TU and the double hash marks indicate that VW XW, but no relationship between the corresponding sides is known. S U V X Problem 2 Sides, Sides, Sides A. What must be true about the sides of similar triangles? Use a complete sentence in your answer. B. In Problem 1, you found that if two pairs of corresponding angles of two triangles are congruent, then the triangles are similar. Suppose that you drew two triangles so that the ratios of two pairs of corresponding sides are equal. Do you think that these triangles would necessarily be similar? Use complete sentences to explain your reasoning. C. Measure the lengths of the sides of the triangle below to the nearest millimeter. Record the lengths on the triangle. M L N 300 Chapter Similarity

22 Problem 2 Sides, Sides, Sides What could be the side lengths of a triangle that is similar to LMN on the previous page? D. Draw a triangle with the side lengths described in part (C). Determine whether this triangle is similar to the triangle in part (C). Explain how you found your answer. Use complete sentences in your answer. Take Note When it is said that the corresponding sides are proportional, it means that the ratios of lengths of the corresponding sides are equal. Investigate Problem 2 1. The result of Problem 2 is called the Side-Side-Side Similarity Postulate. Side-Side-Side Similarity Postulate If the corresponding sides of two triangles are proportional, then the triangles are similar. A B C AB BC AC If then ABC ~ DEF. DE EF DF, D E F Lesson.3 Proving Triangles Similar: AA, SSS, and SAS 301

23 Investigate Problem 2 If the corresponding sides of two triangles are proportional, what makes the triangles similar? Use complete sentences in your answer. 2. Determine whether UVW is similar to XYZ. If so, use symbols to write a similarity statement. Show all your work and use a complete sentence in your answer. 33 meters U V 24 meters W 36 meters 16 meters Y 22 meters Z 24 meters X 3. An art projector has been set up properly, and a piece of art has been projected onto a wall, as shown below. The triangles below are isosceles triangles. Art M L 4 inches 12 inches N Lens 8 inches 24 inches Find the unknown side lengths of LMN and PQN. Show all your work. Leave your answers as radicals in simplest form. Q P Image 302 Chapter Similarity

24 Investigate Problem 2 Show that LMN ~ PQN. Explain your reasoning. Use a complete sentence in your answer. 4. Suppose that you know that the ratios of the lengths of two pairs of corresponding sides of two triangles are equal. How many pairs of corresponding angles do you need to know are congruent in order to determine that the triangles are similar? Which pair(s) of angles would these have to be? Use complete sentences to explain your reasoning. 5. In Question 4, you should have discovered the Side-Angle-Side Similarity Postulate. Side-Angle-Side Similarity Postulate If two of the corresponding sides of two triangles are proportional and the included angles are congruent, then the triangles are similar. B E A C D AB AC If and A D, then ABC ~ DEF. DE DF Can you use this postulate to show that the triangles in Question 3 are similar? If so, explain which angles you would show are congruent and which sides you would show are proportional. Use complete sentences in your answer. F Lesson.3 Proving Triangles Similar: AA, SSS, and SAS 303

25 Investigate Problem 2 6. You can use a paragraph proof to prove that LMN ~ PQN in Question 3. A paragraph proof is a proof that is written in paragraph form. In this kind of proof, you still need to show the logical steps of your argument and give the reasons for the logical steps. Complete the paragraph proof below that proves that LMN ~ PQN. Use the Similarity Postuate. First, find MN and PN by using the. MN MN 2 MN 2 MN PN PN 2 PN 2 PN Then, MN LN and PN QN because the triangles are. Next, find the ratios of the lengths of the corresponding sides. MN PN LN QN So the corresponding sides are. Because MNL and QNP are angles, they are congruent. So by the, LMN ~ PQN. 304 Chapter Similarity

26 .4 Objective In this lesson, you will: Use indirect measurement to find heights and widths of objects. Key Term indirect measurement Modeling a Park Indirect Measurement SCENARIO As part of a science fair project, you are making a fairly accurate model of a local park that is on the edge of a creek and some tall oak trees. To make the model, you need the approximate dimensions of these objects. Because it is not reasonable for you to directly measure the height of a very tall tree, you must come up with a different method. Problem 1 How Tall is That Oak Tree? Use the steps below to experiment with measuring the height of a tall object. You will need a tape measure, a marker, and a mirror, making sure the mirror is absolutely flat. A. Choose an object that you can easily find the height of, such as a short tree or a lamp. Use a marker to make a dot near the center of a mirror. Face the object you would like to measure and place the mirror between yourself and the object. You, the object, and the mirror should form a straight line. Look into the mirror and move directly backward until you can see the top of the object on the dot, as shown below. Place the marker where you are standing. B. Measure the distance between the marker and the dot on the mirror and measure the distance between the dot on the mirror and the object. Record your results on the figure above. C. Measure the height of your eyes and the height of the object and record your results on the figure above. D. Show that the triangles in the figure are similar. Show all your work. Lesson.4 Indirect Measurement 305

27 Problem 1 How Tall is That Oak Tree? E. What do you know about the interior angles of the triangles whose vertices are located at the mirror? Explain your reasoning. Use a complete sentence in your answer. You will find that this relationship between the interior angles of the triangles holds for objects of any height. Investigate Problem 1 1. You go to the park and use the mirror method to gather enough information to find the height of one of the trees. The figure below shows your measurements. Find the height of the tree. Show all your work and use a complete sentence in your answer. 5.5 feet 4 feet 16 feet Take Note Remember that whenever you are solving a problem that involves a kind of measurement like length (or weight) you may have to rewrite some measurements so that they are using the same units. For instance, if a problem involves weight, all of the weights should be measured in grams. 2. A friend wants to try the mirror method on one of the trees. Your friend finds that the distance between her and the mirror is 3 feet and the distance between the mirror and the tree is 18 feet. Your friend s eye height is 60 inches. Draw a diagram of this situation. Then find the height of this tree. Show all your work and use a complete sentence in your answer. 306 Chapter Similarity

28 Investigate Problem 1 3. Your friend notices that the tree is casting a shadow and suggests that you could also use shadows to find the height of the tree. She lines herself up with the tree s shadow so that the tip of her shadow and the tip of the tree s shadow meet. She asks you to measure the distance from the tip of the shadow to her and then measure the distance from her to the tree. You then draw a diagram of this situation as shown below. Find the height of the tree. Show all your work and explain how you found your answer. Use complete sentences in your answer. 15 feet 6 feet 5.5 feet Lesson.4 Indirect Measurement 30

29 Problem 2 How Wide is the Creek? It is not reasonable for you to directly measure the width of a creek, but you can measure the width indirectly. You stand on one side of the creek and your friend stands directly across the creek from you on the other side as shown in the figure. Your friend You A. Your friend is standing 5 feet from the creek and you are standing 5 feet from the creek. Mark these measurements on the diagram above. B. You and your friend walk away from each other in opposite parallel directions. Your friend walks 50 feet and you walk 12 feet. Mark these measurements on the diagram above. Draw a line segment that connects your starting point and ending point and draw a line segment that connects your friend s starting point and ending point. C. Draw a line segment that connects you and your friend s starting points and draw a line segment that connects you and your friend s ending points. Label any angle measures and any angle relationships that you know on the diagram. Use complete sentences to explain how you know these angle measures. D. How do you know that the triangles formed by the lines are similar? Use a complete sentence to explain your reasoning. 308 Chapter Similarity

30 Investigate Problem 2 1. Find the distance from your friend s starting point to your side of the creek. Show all your work and round your answer to the nearest tenth, if necessary. What is the width of the creek? Use complete sentences to explain how you found your answer. 2. There is also a ravine (a deep hollow in the earth) on another edge of the park. You and your friend take measurements like those in Problem 2 to indirectly find the width of the ravine. The figure below shows your measurements. Find the width of the ravine. Show all your work and use a complete sentence in your answer. 15 feet Your friend 6 feet 8 feet You 60 feet Lesson.4 Indirect Measurement 309

31 Investigate Problem 2 3. There is also a large pond in the park. A diagram of the pond is shown below. You want to find the distance across the widest part of the pond, labeled as DE. To indirectly find this distance, you first place a stake at point A. You chose point A so that you can see the edge of the pond on both sides at points D and E, where you also place stakes. Then you tie string from point A to point D and from point A to point E. At a narrow portion of the pond, you place stakes at points B and C along the string so that BC is parallel to DE. The measurements you make are shown on the diagram. Find the distance across the widest part of the pond. Show all your work and use a complete sentence in your answer. 35 feet 16 feet B A 20 feet C D E 310 Chapter Similarity

32 .5 Objectives In this lesson, you will: Find the scale factor of similar solids. Compare the volumes and surface areas of similar solids. Find the dimension of a similar solid given the scale factor. Key Terms similar solids scale factor Making Plastic Containers Similar Solids SCENARIO One way a plastic container can be made is by forcing liquid heated plastic into a mold and injecting air into the mold to form the container. This method is used to make containers in a variety of shapes, such as cylinders and prisms. Problem 1 Comparing Containers Two plastic containers in the shape of rectangular prisms are shown below. 6 inches A 4 inches 4 inches B 6 inches 9 inches 6 inches A. Write and simplify a ratio that compares the length of the base of container A to the length of the base of container B. B. Write a ratio that compares the width of the base of container A to the width of the base of container B. C. Write a ratio that compares the height of container A to the height of container B. D. What do you notice about the ratios? Use a complete sentence in your answer. Lesson.5 Similar Solids 311

33 6 inches A 4 inches 4 inches B 6 inches 9 inches 6 inches Take Note Remember that the volume of a rectangular prism is V Bh where V is the volume, B is the area of the base, and h is the height. The surface area of a rectangular prism is S 2B Ph where S is the surface area, B is the area of the base, P is the perimeter of the base, and h is the height. Investigate Problem 1 1. Just the Math: Similar Solids Containers A and B are similar solids. Two solids with the same shape are similar if the ratios of their corresponding measures (length, width, height, radius) are equal. This ratio is often called the scale factor of one solid to another solid. What is the scale factor of container A to container B? Use a complete sentence in your answer. 2. Find the surface areas of container A and container B. Show all your work and use a complete sentence in your answer. Find the ratio of the surface area of container A to the surface area of container B. Simplify the ratio. Show all your work and use a complete sentence in your answer. 3. Find the volumes of container A and container B. Simplify the ratio. Show all your work and use a complete sentence in your answer. Find the ratio of the volume of container A to the volume of container B. Simplify the ratio. Show all your work and use a complete sentence in your answer. 4. How do the ratios in Question 2 and Question 3 compare to the ratio in Question 1? Use complete sentences in your answer. 312 Chapter Similarity

34 Investigate Problem 1 Why do think that the ratios are related in this way? Use a complete sentence in your answer. 5. The two cylindrical plastic containers below are similar. Complete the following ratios. Write each ratio in simplest form. 84 millimeters 0 millimeters C 2 millimeters D 60 millimeters Radius of cylinder C : Radius of cylinder D Height of cylinder C : Height of cylinder D Take Note What is the scale factor? Use a complete sentence in your answer. Remember that the volume of a cylinder is V r 2 h where V is the volume, r is the radius and h is the height. The surface area of a cylinder is S 2 r 2 2 rh where S is the surface area, r is the radius, and h is the height. Find the surface areas and volumes of the cylinders. Leave your answers in terms of. Complete the ratios. Write each ratio in simplest form. Surface area of cylinder C : Surface area of cylinder D Volume of cylinder C : Volume of cylinder D How does the ratio of the surface areas compare to the scale factor? Use a complete sentence in your answer. Lesson.5 Similar Solids 313

35 Investigate Problem 1 How does the ratio of the volumes compare to the scale factor? Use a complete sentence in your answer. 6. For any two similar solids, how does the ratio of the surface areas compare to the scale factor? Use a complete sentence in your answer. For any two similar solids, how does the ratio of the volumes compare to the scale factor? Use a complete sentence in your answer.. Cube M is similar to cube N with a scale factor of 1 : 4. The length of the base of cube M is 5 inches. Find the length of the base of cube N. What is the ratio of the surface area of cube M to the surface area of cube N? What is the ratio of the volume of cube M to the volume of cube N? Use complete sentences to explain how you found your answers. 314 Chapter Similarity

36 Lesson.5 Similar Solids 315