Similarity 9.1. Expanding Your Mind Dilations of Triangles Look-Alikes. 9.3 Prove It! 9.4 Back on the Grid. Similar Triangles...

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1 Similarity The pupils of a cat's eyes are shaped differently from ours. In brighter light, they appear narrow, like a diamond. But cat's eyes dilate just like ours do. 9.1 Expanding Your Mind Dilations of Triangles Look-Alikes Similar Triangles Prove It! AA, SAS, and SSS Similarity Theorems Back on the Grid Similar Triangles on the Coordinate Plane

2 478 Chapter 9 Similarity

3 Expanding Your Mind Dilations of Triangles Learning Goals In this lesson, you will: Dilate triangles that result in an enlargement of the original triangle. Dilate triangles that result in a reduction of the original triangle. Dilate triangles in a coordinate plane. Key Terms dilation center of dilation scale factor dilation factor enlargement reduction What does it mean if someone says that the pupils of your eyes are dilated? When a light source changes, the pupils of your eyes either shrink or enlarge to control the passage of light. When it is very sunny outside, your pupils will shrink to allow less light in. When it is very dark at night, your pupils will enlarge to allow more light in. A change in light isn t the only thing that makes your pupils dilate. Your pupils can also enlarge when your eyes look at something you like: a favorite show, a cute animal, an interesting picture, or even a special someone. Can you make your fellow pupils pupils dilate? Try it out in your groups. 9.1 Dilations of Triangles 479

4 Problem 1 Maintaining Ratios Enlargements In mathematics, dilations are transformations that produce images that are the same shape as the original image, but not the same size. Each point on the original figure is moved along a straight line and the straight line is drawn from a fixed point known as the center of dilation. The distance each point moves is determined by the scale factor used. The scale factor or dilation factor is the ratio of the distance of the image from the center of dilation to the distance of the original figure from the center of dilation. When the scale factor is greater than one, the image is called an enlargement. A A' P B C B' C' Triangle ABC was dilated to produce triangle A9B9C9 using point P as the center of dilation. Triangle A9B9C9 is an enlargement of triangle ABC. Therefore, the scale factor can be expressed as PA9 PA 5 PB9 5 PC9 PB PC. 1. How is the ratio distance of the image from the center of dilation : distance of the original figure from the center of dilation represented? Is the scale factor less than 1, equal to 1, or greater than 1? Explain your reasoning.. Measure each side of triangle ABC in millimeters. m AB 5 m BC 5 m AC Chapter 9 Similarity

5 3. Measure each side of triangle A9B9C9 in millimeters. m A9B9 5 m B9C9 5 You will need a ruler and a protractor. m A9C Measure each line segment. m A9P 5 mm m AP 5 m B9P 5 mm m BP 5 m C9P 5 mm m CP 5 mm mm mm 5. Determine each ratio. A9P AP 5 C9P CP 5 B9C9 BC 5 B9P BP 5 A9B9 AB 5 A9C AC 5 6. Measure each angle in triangle ABC. m/a 5 m/b 5 m/c 5 7. Measure each angle in triangle A9B9C9. m/a9 5 m/b9 5 m/c Compare triangle A9B9C9 to triangle ABC. What do you notice? Can I now add markers on triangles ABC and A, B, C,? 9.1 Dilations of Triangles 481

6 Problem Maintaining Ratios Reductions When the scale factor or dilation factor is less than one, the image is called a reduction. D' D P E' F' E F Triangle DEF was dilated to produce triangle D9E9F9 using point P as the center of dilation. Triangle D9E9F9 is a reduction of triangle DEF. Therefore, the scale factor can be expressed as PD9 PD 5 PE9 5 PF9 PE PF. 1. How is the ratio distance of the image from the center of dilation : distance of the original figure from the center of dilation represented? Is the scale factor less than 1, equal to 1, or greater than 1? Explain your reasoning.. Measure each side of triangle DEF in millimeters. m DE 5 m EF 5 m DF 5 3. Measure each side of triangle D9E9F9 in millimeters. m D9E9 5 m E9F9 5 m D9F Chapter 9 Similarity

7 4. Measure each line segment. m D9P 5 mm m DP 5 m E9P 5 mm m EP 5 m F9P 5 mm m FP 5 mm mm mm 5. Determine each ratio. D9P DP 5 F9P FP 5 E9F9 EF 5 6. Measure each angle in triangle DEF. m/d 5 m/e 5 E9P EP 5 D9E9 DE 5 D9F DF 5 How do these dilation ratios compare to the dilation ratios from Problem 1? m/f 5 7. Measure each angle in triangle D9E9F9. m/d9 5 m/e9 5 m/f Compare triangle D9E9F9 to triangle DEF. What do you notice? Can I add markers on triangles DEF and D, E, F,? 9.1 Dilations of Triangles 483

8 Problem 3 Dilating Triangles on a Coordinate Plane 1. Enlarge triangle WXY with P as the center of dilation and a scale factor of. Follow the steps given. Take your time and use your straightedge. W P X Y Step 1: Measure PW, PX, and PY in millimeters. m PW 5 m PX 5 Step : m PY 5 Extend line segment PW to point W9 such that m PW9 5 3 m PW. Extend line segment PX to point X9 such that m PX9 5 3 m PX. Extend line segment PY to point Y9 such that m PY9 5 3 m PY. Step 3: Join points W9, X9, and Y9 to form triangle W9X9Y9. How can you verify triangle W, X, Y, was enlarged correctly? 484 Chapter 9 Similarity

9 . Analyze triangle ABC. a. Dilate triangle ABC on the coordinate plane using the origin (0, 0) as the center of dilation and a scale factor of to form triangle A9B9C9. y A (3, 7) 6 4 (3, 3) C B (7, 3) x b. What are the coordinates of points A9, B9, and C9? 3. Graph triangle ABC with the coordinates A(3, 7), B(7, 3), and C(3, 3) on the grid provided. y a. Dilate triangle ABC on the coordinate plane using point C as the center of dilation and a scale factor of 3 to form triangle A9B9C. x b. What are the coordinates of points A9 and B9? 9.1 Dilations of Triangles 485

10 4. Reduce triangle HJK with P as the center of dilation and a scale factor of 1. Follow the steps given. H P J K Step 1: Measure PH, PJ, and PK in millimeters. Step : m PH 5 m PJ 5 m PK 5 Locate point H9 such that m PH m PH. Locate point J9 such that m PJ m PJ. Locate point K9 such that m PK m PK. Step 3: Join points H9, J9, and K9 to form triangle H9J9K9. How can you verify that triangle H, J, K, was reduced correctly? 486 Chapter 9 Similarity

11 5. Analyze triangle ABC. a. Dilate triangle ABC on the coordinate plane using the origin (0, 0) as the center of dilation and a scale factor of 1 to form triangle A9B9C9. y A (, 14) 6 4 C (6, 6) B (14, 6) x b. What are the coordinates of points A9, B9, and C9? 6. Graph triangle ABC with the coordinates A(3, 15), B(15, 3), and C(3, 3) on the grid provided. y a. Dilate triangle ABC on the coordinate plane using point C as the center of dilation and a scale factor of 1 to form triangle A9B9C. x b. What are the coordinates of points A9 and B9? 9.1 Dilations of Triangles 487

12 Talk the Talk In this lesson, several triangles were dilated. Whether it was an enlargement or a reduction, the same conclusions can be drawn about the relationship between corresponding angles and the relationship between the corresponding sides of a triangle and its image resulting from dilation. 1. Describe the relationship between the corresponding angles in an original triangle and its image resulting from dilation.. Describe the relationship between the corresponding sides in an original figure and its image resulting from dilation. 3. Does dilation result in an image that is the same shape as the original? Why or why not? 4. Does dilation result in an image that is the same size as the original? Why or why not? 488 Chapter 9 Similarity

13 5. If two triangles are congruent, what is the relationship between the corresponding angles? 6. If two triangles are congruent, what is the relationship between the corresponding sides? 7. Describe how a triangle is dilated when the ratio distance of the image from the center of dilation : distance of the original figure from the center of dilation is: less than 1. equal to 1. greater than 1. Be prepared to share your solutions and methods. 9.1 Dilations of Triangles 489

14 490 Chapter 9 Similarity

15 Look-Alikes Similar Triangles Learning Goals In this lesson, you will: Key Term similar triangles Define similar triangles. Identify the corresponding parts of similar triangles. Write triangle similarity statements. Determine the measure of corresponding parts of similar triangles. Turquoise, navy, cobalt, robin s egg, cornflower, ultramarine, aquamarine, cerulean, and periwinkle all of these are names for different shades of the color blue. There are an infinite number of possibilities for shades of blue, but all of them are similar in one way: They are all blue. What are some examples of similarity you have learned in mathematics? 9. Similar Triangles 491

16 Problem 1 Similar Triangles Similar triangles are triangles that have the same shape. In the previous lesson, you learned that when a triangle is dilated, the resulting image is an enlarged or reduced triangle that maintains the same shape as the original triangle. Dilations resulted in congruent corresponding angles, and proportional corresponding sides based on the scale factor or dilation ratio. In the figure shown, triangle ABC is similar to triangle A B C. This can be expressed using symbols as ABC, A B C. A' A P C B C' B' 1. Use the figure shown to answer each question. a. Identify the congruent corresponding angles. b. Write ratios to identify the proportional sides. 49 Chapter 9 Similarity

17 . Given TRP, WMY: a. Identify the congruent corresponding angles. b. Write ratios to identify the proportional sides. 3. Suppose /K > /H, /P > /O, /E > /W, and KP 5 PE HO 5 KE OW HW. Write a triangle similarity statement. 9. Similar Triangles 493

18 Problem Unknown Measurements 1. Given: ZAP, EDP PZ 5 5 cm, ZA 5 4 cm, and ED 5 1 cm P 5 cm Z 4 cm A Think about the similar triangles given. What does this tell you? E 1 cm D a. What other measurement(s) can you determine? Explain how you know. b. Determine the measurement(s). 494 Chapter 9 Similarity

19 . Given: ZAP, EDP m/e 5 47 P Z A E 47 o D a. What other measurement(s) can you determine? Explain how you know. b. Determine the measurement(s). 9. Similar Triangles 495

20 3. Given: WRM, WGQ WQ 5 5 cm, WG 5 6 cm, and GR 5 8 cm M Q 5 cm W 6 cm G 8 cm R a. What other measurement(s) can you determine? Explain how you know. b. Determine the measurement(s). 496 Chapter 9 Similarity

21 4. Given: DFH, TKH TK ft, KH 5 6 ft, and FK 5 15 ft D T 5.5 ft F 15 ft K 6 ft H a. What other measurement(s) can you determine? Explain how you know. b. Determine the measurement(s). 9. Similar Triangles 497

22 5. Given: WBE, SEP BP 5 EP, WE 5 58 mm W S B P E a. What other measurement(s) can you determine? Explain how you know. b. Determine the measurement(s). Be prepared to share your solutions and methods. 498 Chapter 9 Similarity

23 Prove It! AA, SAS, and SSS Similarity Theorems Learning Goals In this lesson, you will: Explore the AA Similarity Theorem. Explore the SAS Similarity Theorem. Explore the SSS Similarity Theorem. Use the AA, SAS, and SSS Similarity Theorems to identify similar triangles. Key Terms AA Similarity Theorem SAS Similarity Theorem SSS Similarity Theorem Graphic artists often use knowledge about similarity to create realistic-looking perspective drawings. Choose where the horizon should be and a vanishing point a point where all parallel lines in the drawing should appear to meet and you too can create a perspective drawing. Can you see how similarity was used to create this drawing? Can you use similarity to create your own perspective drawing? vanishing point horizon 9.3 AA, SAS, and SSS Similarity Theorems 499

24 Problem 1 Two Angles In the previous lesson, you determined that when two triangles are similar, the corresponding angles are congruent and the corresponding sides are proportional. To show that two triangles are similar, do you need to show that all of the corresponding sides are proportional and all of the corresponding angles are congruent? In this lesson, you will explore efficient methods for showing that two triangles are similar. 1. If the measures of two angles of a triangle are known, is that enough information to draw a similar triangle? Let s explore this possibility. a. Use a straightedge to draw triangle ABC in the space provided. b. Use a protractor to measure, /A and /B, of triangle ABC and record the measurements. m/a 5 m/b 5 c. Do you need a protractor to determine m/c? Why or why not? Do you remember what the sum of the angle measures in a triangle is? d. Use the measurements in part (b) to draw triangle DEF in the space provided. 500 Chapter 9 Similarity

25 e. Based on your knowledge from the previous lesson, what other information is needed to determine if the two triangles are similar and how can you acquire that information? f. Determine the measurements to get the additional information needed and decide if the two triangles are similar. You have just shown that given the measures of two pairs of congruent corresponding angles of two triangles, it is possible to determine that two triangles are similar. In the study of geometry, this is expressed as a theorem. The Angle-Angle (AA) Similarity Theorem states that if two angles of one triangle are congruent to the corresponding angles of another triangle, then the triangles are similar. 9.3 AA, SAS, and SSS Similarity Theorems 501

26 . Analyze triangle ABC. a. Dilate triangle ABC on the coordinate plane using the origin (0, 0) as the center of dilation and a scale factor of 3 to form triangle A9B9C9. y A (3, 7) 6 4 (3, 3) C B (7, 3) x b. What are the coordinates of points A9, B9, and C9? c. Use the AA Similarity Theorem and a protractor to determine if the original triangle, ABC, and the image resulting from the dilation, A9B9C9, are similar triangles. If the center of dilation is at the origin, can that help you determine the coordinates of A,, B,, and C,? 50 Chapter 9 Similarity

27 Problem Two Sides and the Included Angle If the lengths of two sides and the measure of the included angle of a triangle are known, is that enough information to draw a similar triangle? 1. Let s explore this possibility. a. Use a straightedge to draw triangle ABC in the space provided. Remember, you explored a similar situation when analyzing congruent triangles. b. Use a ruler to measure the lengths of AB and BC, of triangle ABC and record the measurements. m AB 5 m BC 5 c. Use a protractor to measure /B, the included angle in triangle ABC, and record the measurement. m/b 5 d. Use the measurements in parts (b) to draw two sides of a triangle that are proportional to the corresponding sides of triangle ABC, and use the angle measure in part (c) to draw an included angle that is congruent, in order to form triangle DEF in the space provided. 9.3 AA, SAS, and SSS Similarity Theorems 503

28 e. Based on your knowledge from the previous lesson, what other information is needed to determine if the two triangles are similar and how can you acquire that information? f. Determine the measurements to get the additional information needed and decide if the two triangles are similar. You have just shown that given the lengths of two sides of a triangle and the measure of the included angle, it is possible to determine that two triangles are similar. In the study of geometry, this is expressed as a theorem. The Side-Angle-Side (SAS) Similarity Theorem states that if two pairs of corresponding sides of two triangles are proportional and the included angles are congruent, then the triangles are similar. 504 Chapter 9 Similarity

29 . Use the SAS Similarity Theorem and a protractor to determine if the two triangles drawn on the coordinate plane are similar. Use a protractor to verify the measure of the included angle. y (, 6) R N (3.5, 4) P (7, 6) x A (1, 8) (, 1) 10 M 1 (8, 1) Q 9.3 AA, SAS, and SSS Similarity Theorems 505

30 Problem 3 Three Sides If the lengths of three sides of a triangle are known, is that enough information to draw a similar triangle? 1. Let s explore this possibility. a. Use a straightedge to draw triangle ABC in the space provided. Remember, you explored a similar situation with congruent triangles. b. Use a ruler to measure the length of each side, AB, BC, and AC, of triangle ABC and record the measurements. m AB 5 m BC 5 m AC 5 c. Use the measurements in parts (b) to draw three sides of a triangle that are proportional to these measurements to form triangle DEF in the space provided. d. Michael says that based on what he s learned so far, he needs to find the measures of the three corresponding angles of the triangles to determine if they are similar. Is he correct? Why or why not? e. Determine the measurements to get the additional information needed and decide if the two triangles are similar. 506 Chapter 9 Similarity

31 You have just shown that given the length of three sides of a triangle, it is possible to determine that two triangles are similar. In the study of geometry, this method is expressed as a theorem. The Side-Side-Side (SSS) Similarity Theorem states that if three pairs of corresponding sides of two triangles are proportional, then the triangles are similar.. Use the SSS Similarity Theorem to determine if the two triangles drawn on the coordinate plane are similar. y 1 (6, 10) M 10 8 (5.5, 7) Q 6 4 (4, 1) R P A (9, ) (3, ) 4 6 N (7, 1) x AA, SAS, and SSS Similarity Theorems 507

32 Talk the Talk Determine if each pair of triangles are similar by AA, SAS, or SSS. 1. J 35 cm K M 30 cm 6 cm E R 7 cm T. W X 31 o V 31 o Z Y 3. N 9 o S M 9 o R T 4. P 4 mm 5 mm 5 mm F Q 6 mm 13.5 mm R 6.5 mm G Be prepared to share your solutions and methods. 508 Chapter 9 Similarity

33 Back on the Grid Similar Triangles on the Coordinate Plane Learning Goals In this lesson, you will: Graph similar triangles and determine the dilation factor. Dilate triangles to form similar triangles. Verify that triangles are similar using Similarity Theorems. Determine the coordinates of a point needed to form similar triangles. You know that the Earth is farther away from the Sun than the planet Venus. But is there a way to tell just by looking? Amazingly, the ancient Greeks used similar triangles to figure this out. They observed that Venus was never more than 47 high in the sky at sunset. Venus 47 o Sun Earth And they knew something that you will learn later in mathematics that for any angle in a right triangle, the ratio of the opposite side length to the hypotenuse is always the same for all similar triangles, no matter what these lengths are. The ancient Greeks knew that for a 47 angle, the ratio of the opposite side to the hypotenuse was about 0.7. This meant that the Venus Sun distance was about 7 10 of the Earth Sun distance. Venus is closer! Can you see how they did it? 9.4 Similar Triangles on the Coordinate Plane 509

34 Problem 1 Similar Triangles Resulting from Dilations Let s explore methods for showing two triangles are similar through dilations. 1. Triangle ABC has vertices A(3, 3), B(8, 3), and C(8, 3). a. Graph triangle ABC. y x b. Triangle DEF is the image that resulted from the dilation of triangle ABC. The coordinates of triangle DEF are D (1.5, 1.5), E (4, 1.5), and F (4, 1.5). Graph triangle DEF on the coordinate plane in part (a). c. What scale factor was used? d. How did you determine the scale factor? e. Is the dilation an enlargement or reduction? Explain your reasoning. 510 Chapter 9 Similarity

35 f. How can you verify that triangle ABC and triangle DEF are similar? g. Use the SAS Similarity Theorem to verify triangle ABC is similar to triangle DEF. 9.4 Similar Triangles on the Coordinate Plane 511

36 . Triangle MAP is the image that resulted from the dilation of triangle QRN. y (5, 7) Q R (7, 7) P (14, ) N (7, 1) 8 10 x A (14, 14) M (10, 14) 14 a. What scale factor was used? b. How did you determine the scale factor? c. Is the dilation an enlargement or reduction? Explain your reasoning. d. How can you verify that triangle MAP and triangle QRN are similar? 51 Chapter 9 Similarity

37 e. Use the SSS Similarity Theorem to verify triangle MAP is similar to triangle QRN. 9.4 Similar Triangles on the Coordinate Plane 513

38 3. Analyze triangle PWN. a. Dilate triangle PWN shown using a scale factor of 4 to form triangle GKA. y W (3, 3) ( 1, 1) P N (3, 0) x b. What are the coordinates of the dilated image? c. How did you determine the coordinates of the dilated image? d. Is the dilation an enlargement or reduction? Explain your reasoning. e. How can you verify that triangle PWN and triangle GKA are similar? f. Use the AA Similarity Theorem to verify triangle PWN is similar to triangle GKA. 514 Chapter 9 Similarity

39 4. Analyze triangle ZEN. a. Dilate triangle ZEN shown using a scale factor of 1 to form triangle FRB. y E (8, 8) 1 10 Z (5, 3) N (10, 1) 10 1 x b. What are the coordinates of the dilated image? c. How did you determine the coordinates of the dilated image? d. Is the dilation an enlargement or reduction? Explain your reasoning. e. Use the AA Similarity Theorem to verify triangle ZEN is similar to triangle FRB. 9.4 Similar Triangles on the Coordinate Plane 515

40 Talk the Talk This chapter focused on three methods to show that two triangles are similar. Complete the graphic organizer by listing the methods and provide an illustration of each method. Be prepared to share your solutions and methods. 516 Chapter 9 Similarity

41 Chapter 9 Summary Key Terms dilation (9.1) center of dilation (9.1) scale factor (9.1) dilation factor (9.1) enlargement (9.1) reduction (9.1) similar triangles (9.) AA Similarity Theorem (9.3) SAS Similarity Theorem (9.3) SSS Similarity Theorem (9.3) Dilating Triangles Dilations are transformations that produce images that are the same shape as the original image, but not the same size. Each point on the original figure is moved along a straight line and the straight line is drawn from a fixed point known as the center of dilation. The scale factor is the ratio formed when comparing the distance of the image from the center of dilation to the distance of the original figure from the center of dilation. Example Enlarge triangle ABC with P as the center of dilation and a scale factor of. B B P C A C A First, measure PA. Then, extend the line PA to the point A9 such that PA9 5 PA. Next, measure PB. Then, extend the line PB to the point B9 such that PB9 5 PB. Finally, measure PC. Then, extend the line PC to the point C9 such that PC9 5 PC. Because the scale factor is greater than one, the image A9B9C9 is called an enlargement. If the scale factor had been less than one, the image would be called a reduction. Chapter 9 Summary 517

42 Properties of Similar Triangles Similar triangles are triangles that have the same shape. Example In the figure shown, triangle DEF is similar to triangle D9E9F9. This can be expressed using symbols as ndef nd9e9f9. P E E D F D 1. Identify the congruent corresponding angles. /D > /D9 /E > /E9 /F > /F9. Write ratios to identify the proportional sides. D9E9 5 E9F9 DE 5 F9D9 EF FD F I am going to be an architect and they use these skills all the time. Understanding this already will be a big help when it comes time to go to college. 518 Chapter 9 Similarity

43 Using Similar Triangles to Find Unknown Measures The properties of similar triangles can be used to determine unknown measures of the triangles. Example In the figure shown, triangle ACE is similar to triangle BCD. Find the length of side AC. C 9 cm B 6 cm D A 1 cm E Using the proportional relationship between corresponding sides of similar triangles, you know BC 5 BD AC AE. Substitute the known values for the sides. BC 5 BD AC AE AC 1 6AC 5 (9)(1) 6AC 5108 AC 518 The length of side AC is 18 centimeters. Chapter 9 Summary 519

44 AA, SAS, and SSS Similarity Theorems The Angle-Angle (AA) Similarity Theorem states: If two angles of one triangle are congruent to the corresponding angles of another triangle, then the triangles are similar. The Side-Angle-Side (SAS) Similarity Theorem states: If two pairs of corresponding sides of two triangles are proportional and the included angles are congruent, then the triangles are similar. The Side-Side-Side (SSS) Similarity Theorem states: If three pairs of corresponding sides of two triangles are proportional, then the triangles are similar. Example Determine if each pair of triangles are similar by AA, SAS, or SSS. 1. B /A 5 /F /B 5 /D nabc nfde 36 E 7 F The triangles are similar by AA. 36 D 7 A C. C AB AC 5 AE AD 3 cm B 6 cm /A 5 /A nabe nacd The triangles are similar by SAS. A 6 cm E 3 cm D 50 Chapter 9 Similarity

45 3. B AB 5 14 DE 7 BC 5 1 EF 6 14 cm E 1 cm 6 cm 7 cm 7 cm F C CA 5 14 FD AB DE 5 BC 5 CA EF FD A D 14 cm nabc ndef The triangles are similar by SSS. Similar Triangles on the Coordinate Plane Properties of similar triangles can be used to graph similar triangles, determine the scale factor that was used to create the triangles, and verify that the triangles are similar. Example Triangle ABC has vertices A(1,), B(1, 5), C(5, ). Graph triangle ABC E B 4 A 0 0 D F C Chapter 9 Summary 51

46 Triangle DEF is the image that resulted from the dilation of triangle ABC. The coordinates of triangle DEF are D(3,6), E(3, 15), F(15, 6). Graph triangle DEF. The scale factor used was 3. Use the SAS Similarity Theorem to verify triangle ABC is similar to triangle DEF. /A is congruent to /D because they are both right angles and all right angles are congruent. AB 5 AC DE DF The lengths of the sides that include /A and /D are proportional. Triangle ABC is similar to triangle DEF by the SAS Similarity Theorem. 5 Chapter 9 Similarity

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