Big and Small. Dilating Triangles to Create Similar Triangles. Lesson 4.1 Skills Practice. Vocabulary. Problem Set. Define the term in your own words.
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1 Lesson.1 Skills Practice Name Date Big and Small Dilating Triangles to Create Similar Triangles Vocabulary Define the term in your own words. 1. similar triangles Problem Set Rectangle L9M9N9P9 is a dilation of rectangle LMNP. The center of dilation is point Z. Use a metric ruler to determine the actual lengths of ZL, ZN, ZM, ZP, ZL9, ZN9, ZM9, and ZP9 to the nearest tenth of a centimeter. Then, express the ratios ZL9 ZL, ZN9 ZN, ZM9 ZM, and ZP9 as decimals. ZP 1. N' P' P N Z L M M' L' ZL 5 2, ZN 5 3, ZM 5 3, ZP 5 2, ZL9 5 5, ZN , ZM , ZP ZL9 ZL , ZN9 ZN , ZM9 3 ZM , ZP9 3 ZP Chapter Skills Practice 65
2 Lesson.1 Skills Practice page 2 2. L M L' P M' P' N Z N' 3. M' N' L' M N P' L Z P 66 Chapter Skills Practice
3 Lesson.1 Skills Practice page 3 Name Date. Z L' M' L P' N' M P N Chapter Skills Practice 67
4 Lesson.1 Skills Practice page Given the image and pre-image, determine the scale factor. 5. y A A C B C B x The scale factor is 2. Each coordinate of the image is two times the corresponding coordinate of the pre-image. ABC has vertex coordinates A(2, 3), B(5, 1), and C(2, 1). A9B9C9 has vertex coordinates A9(, 6), B9(10, 2), and C9(, 2). 6. y 16 D D E F 2 E F x 68 Chapter Skills Practice
5 Lesson.1 Skills Practice page 5 Name Date 7. y 16 G G H I 2 H I x 8. y L L J K J K x Chapter Skills Practice 69
6 Lesson.1 Skills Practice page 6 Given the pre-image, scale factor, and center of dilation, use a compass and straight edge to graph the image. 9. The scale factor is 3 and the center of dilation is the origin. y B B A C A C x 10. The scale factor is and the center of dilation is the origin. y A 2 0 C B x 11. The scale factor is 2 and the center of dilation is the origin. y F E D x 70 Chapter Skills Practice
7 Lesson.1 Skills Practice page 7 Name Date 12. The scale factor is 5 and the center of dilation is the origin. y E 0 D F x Use coordinate notation to determine the coordinates of the image. 13. ABC has vertices A(1, 2), B(3, 6), and C(9, 7). What are the vertices of the image after a dilation with a scale factor of using the origin as the center of dilation? A(1, 2) A9((1), (2)) 5 A9(, 8) B(3, 6) B9((3), (6)) 5 B9(12, 2) C(9, 7) C9((9), (7)) 5 C9(36, 28) 1. DEF has vertices D(8, ), E(2, 6), and F(3, 1). What are the vertices of the image after a dilation with a scale factor of 5 using the origin as the center of dilation? 15. GHI has vertices G(0, 5), H(, 2), and I(3, 3). What are the vertices of the image after a dilation with a scale factor of 9 using the origin as the center of dilation? 16. JKL has vertices J(6, 2), K(1, 3), and L(7, 0). What are the vertices of the image after a dilation with a scale factor of 12 using the origin as the center of dilation? Chapter Skills Practice 71
8 Lesson.1 Skills Practice page ABC has vertices A(8, ), B(1, 16), and C(6, 10). What are the vertices of the image after a dilation with a scale factor of 1 using the origin as the center of dilation? DEF has vertices D(25, 25), E(15, 10), and F(20, 10). What are the vertices of the image after a dilation with a scale factor of 1 using the origin as the center of dilation? GHI has vertices G(0, 20), H(16, 2), and I(12, 12). What are the vertices of the image after a dilation with a scale factor of 3 using the origin as the center of dilation? 20. JKL has vertices J(8, 2), K(6, 0), and L(, 10). What are the vertices of the image after a dilation with a scale factor of 5 using the origin as the center of dilation? 2 72 Chapter Skills Practice
9 Lesson.2 Skills Practice Name Date Similar Triangles or Not? Similar Triangle Theorems Vocabulary Give an example of each term. Include a sketch with each example. 1. Angle-Angle Similarity Theorem 2. Side-Side-Side Similarity Theorem Chapter Skills Practice 73
10 Lesson.2 Skills Practice page 2 3. Side-Angle-Side Similarity Theorem. included angle 5. included side 7 Chapter Skills Practice
11 Lesson.2 Skills Practice page 3 Name Date Problem Set Explain how you know that the triangles are similar. 1. The triangles are congruent by the Angle-Angle Similarity Theorem. Two corresponding angles are congruent ft 3 ft.5 ft.5 ft 2 ft 3 ft Chapter Skills Practice 75
12 Lesson.2 Skills Practice page. in. in. 8 in. 8 in. 5 in. 10 in cm 60 3 cm 9 cm 60 6 cm 6. 5 mm 10 mm 0 7 mm 1 mm 0 76 Chapter Skills Practice
13 Lesson.2 Skills Practice page 5 Name Date Determine what additional information you would need to prove that the triangles are similar using the given theorem. 7. What information would you need to use the Angle-Angle Similarity Theorem to prove that the triangles are similar? To prove that the triangles are similar using the Angle-Angle Similarity Theorem, the first triangle should have a corresponding 60 degree angle and the second triangle should have a corresponding 35 degree angle. 8. What information would you need to use the Angle-Angle Similarity Theorem to prove that the triangles are similar? What information would you need to use the Side-Angle-Side Similarity Theorem to prove that the triangles are similar? 6 m m 10 m Chapter Skills Practice 77
14 Lesson.2 Skills Practice page What information would you need to use the Side-Angle-Side Similarity Theorem to prove that the triangles are similar? 5 in. 5 in. 9 in. 9 in. 11. What information would you need to use the Side-Side-Side Similarity Theorem to prove that these triangles are similar? 12 cm 1 cm 6 cm 5 cm 12. What information would you need to use the Side-Side-Side Similarity Theorem to prove that these triangles are similar? ft 6 ft 78 Chapter Skills Practice
15 Lesson.2 Skills Practice page 7 Name Date Determine whether each pair of triangles is similar. Explain your reasoning. 13. Y 10 yd T 10.8 yd R 6 yd X 18 yd Z S The triangles are similar by the Side-Angle-Side Similarity Theorem because the included angles in both triangles are congruent and the corresponding sides are proportional. XY RS XZ RT K B 9 cm 6 cm 8 cm 12 cm J 12 cm L A 16 cm C Chapter Skills Practice 79
16 Lesson.2 Skills Practice page N 60 3 in. 2 in. M O Q in. 5 in. R P 16. P 100 R 20 T Q 100 S 60 U 80 Chapter Skills Practice
17 Lesson.2 Skills Practice page 9 Name Date 17. V 10 m W 18 m X A.5 m B 2.5 m C 18. K 3 ft 5 ft J 6 ft L I 10 ft 12 ft H 7 ft G Chapter Skills Practice 81
18 Lesson.2 Skills Practice page M O N P R Q 20. D 15 m 18 m E 27 m F S 36 m 2 m T 20 m U 82 Chapter Skills Practice
19 Lesson.3 Skills Practice Name Date Keep It in Proportion Theorems About Proportionality Vocabulary Match each definition to its corresponding term. 1. Angle Bisector/Proportional Side Theorem a. If a line parallel to one side of a triangle intersects the other two sides, then it divides the two sides proportionally. 2. Triangle Proportionality Theorem b. A bisector of an angle in a triangle divides the opposite side into two segments whose lengths are in the same ratio as the lengths of the sides adjacent to the angle. 3. Converse of the Triangle c. If a line divides two sides of a triangle proportionally, Proportionality Theorem then it is parallel to the third side.. Proportional Segments Theorem d. The midsegment of a triangle is parallel to the third side of the triangle and half the measure of the third side of the triangle 5. Triangle Midsegment Theorem e. If three parallel lines intersect two transversals, then they divide the transversals proportionally. Chapter Skills Practice 83
20 Lesson.3 Skills Practice page 2 Problem Set Calculate the length of the indicated segment in each figure. 1. HJ bisects H. Calculate HF. 2. LN bisects L. Calculate NM. F 15 cm J 18 cm G 8 in. L 5 in. H 21 cm K in. N M The length of segment HF is 17.5 centimeters. GH 5 GJ HF JF HF 15 18? HF HF BD bisects B. Calculate AD.. SQ bisects S. Calculate SP. B 3 ft C 2 ft P 9 m Q 12 m 6 ft D S 18 m R A 8 Chapter Skills Practice
21 Lesson.3 Skills Practice page 3 Name Date 5. YZ bisects Y. Calculate YW. 6. VX bisects V. Calculate XW. cm Y U 9 ft V X 8 cm Z 9 cm W 10 ft X 6 ft W 7. GE bisects G. Calculate FD. D 8. ML bisects M. Calculate NL. N 5 cm 15 cm E L 11 mm G 18 cm F K 5 mm mm M Chapter Skills Practice 85
22 Lesson.3 Skills Practice page Use the given information to answer each question. 9. On the map shown, Willow Street bisects the angle formed by Maple Avenue and South Street. Mia s house is 5 miles from the school and miles from the fruit market. Rick s house is 6 miles from the fruit market. How far is Rick s house from the school? Mia s house Willow Street Maple Avenue Fruit market School South Street Rick s house River Avenue Rick s house is 7.5 miles from the school. 5 5 x 6 x 5 30 x Jimmy is hitting a golf ball towards the hole. The line from Jimmy to the hole bisects the angle formed by the lines from Jimmy to the oak tree and from Jimmy to the sand trap. The oak tree is 200 yards from Jimmy, the sand trap is 320 yards from Jimmy, and the hole is 250 yards from the sand trap. How far is the hole from the oak tree? Jimmy Oak tree Hole Sand trap 86 Chapter Skills Practice
23 Lesson.3 Skills Practice page 5 Name Date 11. The road from Central City on the map shown bisects the angle formed by the roads from Central City to Minville and from Central City to Oceanview. Central City is 12 miles from Oceanview, Minville is 6 miles from the beach, and Oceanview is 8 miles from the beach. How far is Central City from Minville? Minville Beach Oceanview Central City 12. Luigi is racing a remote control car from the starting point to the winner s circle.that path bisects the angle formed by the lines from the starting point to the house and from the starting point to the retention pond. The house and the retention pond are each 500 feet from the starting point. The house is 720 feet from the retention pond. How far is the winner s circle from the retention pond? Starting point House Winner s circle Retention pond Chapter Skills Practice 87
24 Lesson.3 Skills Practice page 6 Use the diagram and given information to write a statement that can be justified using the Proportional Segments Theorem, Triangle Proportionality Theorem, or its Converse. State the theorem used. 13. A 1. B D E B C Q A P AD 5 AE, Triangle Proportionality Theorem DB EC C 15. A L 1 L 2 y x a b L 3 D 6 E 8 C B 17. A 18. E B P Q C F 3 cm P 3.9 cm 2. cm Q 3.6 cm D 88 Chapter Skills Practice
25 Lesson.3 Skills Practice page 7 Name Date 19. E 2.5 F I 20. G H G H F A B C D E Use the Triangle Proportionality Theorem and the Proportional Segments Theorem to determine the missing value. 21. Q 2 S P 6 R x x T 6 6x x x 5 3 Chapter Skills Practice 89
26 Lesson.3 Skills Practice page A. x 3 8x D E 6.3 3x 1 5x 3 x B C Use the diagram and given information to write two statements that can be justified using the Triangle Midsegment Theorem. 25. B 26. R D E V W A C T S Given: RST is a triangle D is the midpoint of AB V is the midpoint of RT E is the midpoint of BC W is the midpoint of RS AC, DE 5 2 AC Given: ABC is a triangle DE i 90 Chapter Skills Practice
27 Lesson.3 Skills Practice page 9 Name Date 27. M Y 28. X N X P Given: MNP is a triangle Given: XYZ is a triangle X is the midpoint of MP T is the midpoint of YZ Y is the midpoint of MN U is the midpoint of XY Z T Y U Given each diagram, compare the measures described. Simplify your answers, but do not evaluate any radicals. 29. The sides of triangle LMN have midpoints O(0, 0), P(29, 23), and Q(0, 23). Compare the length of OP to the length of LM. y 8 6 P N O 0 Q 2 6 M 8 x L 6 8 Segment LM is two times the length of segment OP. OP 5 (29 2 0) 2 1 (23 2 0) 2 5 (29) 2 1 (23) LM 5 (10 2 (28) 2 1 (0 2 (26)) Chapter Skills Practice 91
28 Lesson.3 Skills Practice page The sides of triangle LMN have midpoint P(0, 0), Q(22, 2), and R(2, 2). Compare the length of QP to the length of LN. y L ( 2, 2) 3 (2, 2) Q 2 R M P 1 (0, 0) 2 3 x N Chapter Skills Practice
29 Lesson. Skills Practice Name Date Geometric Mean More Similar Triangles Vocabulary Write the term from the box that best completes each statement. Right Triangle/Altitude Similarity Theorem Right Triangle Altitude/Hypotenuse Theorem geometric mean Right Triangle Altitude/Leg Theorem 1. The states that if an altitude is drawn to the hypotenuse of a right triangle, then the two triangles formed are similar to the original triangle and to each other. 2. The states that if the altitude is drawn to the hypotenuse of a right triangle, each leg of the right triangle is the geometric mean of the measure of the hypotenuse and the measure of the segment of the hypotenuse adjacent to the leg. 3. The of two positive numbers a and b is the positive number x such that a x 5 x b.. The states that the measure of the altitude drawn from the vertex of the right angle of a right triangle to its hypotenuse is the geometric mean between the measures of the two segments of the hypotenuse. Problem Set Construct an altitude to the hypotenuse of each right triangle. 1. K 2. J L M O N Chapter Skills Practice 93
30 Lesson. Skills Practice page U Q S R T V Use each similarity statement to write the corresponding sides of the triangles as proportions. 5. CGJ, MKP 6. XZC, YMN CG MK 5 GJ KP 5 CJ MP 7. ADF, GLM 8. WNY, CQR Use the Right Triangle/Altitude Similarity Theorem to write three similarity statements involving the triangles in each diagram. 9. H 10. M P R Q G P Z HPG, PQG, HPG, HQP, 11. PQG, HQP N K L T 12. W U M N 9 Chapter Skills Practice
31 Lesson. Skills Practice page 3 Name Date Solve for x. 13. R x B 20 cm T 1. D 8 cm x A 6 in. F 6 in. G F FB TB 5 RB FB 8 5 x x 5 6 x cm 15. M x Z 16. R 9 in. S cm 12 cm 6 in. Q N P x V Chapter Skills Practice 95
32 Lesson. Skills Practice page 17. B m D T 18. G K x mi L 25 mi 10 m x F L Solve for x, y, and z. 19. W y x z C 25 P H CP 5 WP WP PH x y x 2 5 z 2 25 x 5 x y z 2 x y z 2 x y z 96 Chapter Skills Practice
33 Lesson. Skills Practice page 5 Name Date 20. B 3 V x y 15 R z Y 21. K z 8 y L N x P Chapter Skills Practice 97
34 Lesson. Skills Practice page K H 5 x y A z T Use the given information to answer each question. 23. You are on a fishing trip with your friends. The diagram shows the location of the river, fishing hole, camp site, and bait store. The camp site is located 200 feet from the fishing hole. The bait store is located 110 feet from the fishing hole. How wide is the river? Bait store Camp site Fishing hole River The river is 60.5 feet wide x 200x 5 12,100 x Chapter Skills Practice
35 Lesson. Skills Practice page 7 Name Date 2. You are standing at point D in the diagram looking across a bog at point B. Point B is 8 yards from point A and 189 yards from point C. How wide across is the bog? A B Bog D C 25. Marsha wants to walk from the parking lot through the forest to the clearing, as shown in the diagram. She knows that the forest ranger station is 15 feet from the flag pole and the flag pole is 350 feet from the clearing. How far is the parking lot from the clearing? Clearing Forest ranger station Flag pole Parking lot Forest Chapter Skills Practice 99
36 Lesson. Skills Practice page Andre is camping with his uncle at one edge of a ravine. The diagram shows the location of their tent. The tent is 1.2 miles from the fallen log and the fallen log is 0.75 miles from the observation tower. How wide is the ravine? Tent Ravine Observation tower Fallen log 500 Chapter Skills Practice
37 Lesson.5 Skills Practice Name Date Proving the Pythagorean Theorem Proving the Pythagorean Theorem and the Converse of the Pythagorean Theorem Problem Set Write the Given and Prove statements that should be used to prove the indicated theorem using each diagram. 1. Prove the Pythagorean Theorem. 2. Prove the Converse of the Pythagorean Theorem. C D B A Given: ABC with right angle C Prove: AC 2 1 BC 2 5 AB 2 F Given: Prove: E 3. Prove the Pythagorean Theorem.. Prove the Pythagorean Theorem. X v W w Q w x x v V x v w Given: x v w P S Given: Prove: Prove: Chapter Skills Practice 501
38 Lesson.5 Skills Practice page 2 5. Prove the Pythagorean Theorem. 6. Prove the Converse of the Pythagorean Theorem. G h F U g f g t s h f f g H h S u T g f g h Given: Given: Prove: Prove: Prove each theorem for the diagram that is given. Use the method indicated. In some cases, the proof has been started for you. 7. Prove the Pythagorean Theorem using similar triangles. Draw an altitude to the hypotenuse at point N. K N L According to the Right Triangle/Similarity Theorem, KLM, LMN, MKN. According to the definition of similar triangles, the sides of similar triangles are proportional. KL 5 KM KM KN KM 2 5 KL 3 KN KL 5 ML ML NL ML 2 5 KL 3 NL M KM 2 1 ML 2 5 KL 3 KN 1 KL 3 NL KM 2 1 ML 2 5 KL(KN 1 NL) KM 2 1 ML 2 5 KL(KL) KM 2 1 ML 2 5 KL Chapter Skills Practice
39 Lesson.5 Skills Practice page 3 Name Date 8. Prove the Pythagorean Theorem using algebraic reasoning. n m m s s n n s s m m n 9. Prove the Converse of the Pythagorean Theorem using algebraic reasoning and the SSS Theorem. Construct a right triangle that has legs the same R lengths as the original triangle, q and r, opposite angles Q and R. Label the hypotenuse of this triangle t, opposite the right angle T. q s S r Q Chapter Skills Practice 503
40 Lesson.5 Skills Practice page 10. Prove the Pythagorean Theorem using similarity. h m k m h k k m k h m k m h 11. Prove the Converse of the Pythagorean Theorem using algebraic reasoning. Construct a right triangle that shares side h with the original triangle and has one leg length g, as in the original triangle, and the other leg length w. H f G g h g F w 50 Chapter Skills Practice
41 Lesson.5 Skills Practice page 5 Name Date 12. Prove the Pythagorean Theorem using similar triangles. Given: ABC with right angle C Place side a along the diameter of a circle of radius c so that B is at the center of the circle. b A c D c a C a B c E Chapter Skills Practice 505
42 506 Chapter Skills Practice
43 Lesson.6 Skills Practice Name Date Indirect Measurement Application of Similar Triangles Vocabulary Provide an example of the term. 1. indirect measurement Problem Set Explain how you know that each pair of triangles are similar. 1. A B C D E The angles where the vertices of the triangle intersect are vertical angles, so angles /ACB and /ECD are congruent. The angles formed by BD intersecting the two parallel lines are right angles, so they are also congruent. So by the Angle-Angle Similarity Theorem, the triangles formed are similar. Chapter Skills Practice 507
44 Lesson.6 Skills Practice page 2 2. F G H I J 3. 6 in. 3 in. 8 in. in.. B D A C E 508 Chapter Skills Practice
45 Lesson.6 Skills Practice page 3 Name Date Use indirect measurement to calculate the missing distance. 5. Elly and Jeff are on opposite sides of a canyon that runs east to west, according to the graphic. They want to know how wide the canyon is. Each person stands 10 feet from the edge. Then, Elly walks 2 feet west, and Jeff walks 360 feet east. 2 ft 10 ft 10 ft 360 ft What is the width of the canyon? The distance across the canyon is 10 feet x x x Zoe and Ramon are hiking on a glacier. They become separated by a crevasse running east to west. Each person stands 9 feet from the edge. Then, Zoe walks 8 feet east, and Ramon walks 12 feet west. 9 ft 8 ft 9 ft 12 ft What is the width of the crevasse? Chapter Skills Practice 509
46 Lesson.6 Skills Practice page 7. Minh wanted to measure the height of a statue. She lined herself up with the statue s shadow so that the tip of her shadow met the tip of the statue s shadow. She marked the spot where she was standing. Then, she measured the distance from where she was standing to the tip of the shadow, and from the statue to the tip of the shadow. 5 ft Minh 8 ft 12 ft What is the height of the statue? 8. Dimitri wants to measure the height of a palm tree. He lines himself up with the palm tree s shadow so that the tip of his shadow meets the tip of the palm tree s shadow. Then, he asks a friend to measure the distance from where he was standing to the tip of his shadow and the distance from the palm tree to the tip of its shadow. Dimitri 6 ft 5 ft ft What is the height of the palm tree? 510 Chapter Skills Practice
47 Lesson.6 Skills Practice page 5 Name Date 9. Andre is making a map of a state park. He finds a small bog, and he wants to measure the distance across the widest part. He first marks the points A, C, and E. Andre measures the distances shown on the image. Andre also marks point B along AC and point D along AE, such that BD is parallel to CE. A B 1 ft D 12 ft 126 ft C E What is the width of the bog at the widest point? 10. Shira finds a tidal pool while walking on the beach. She wants to know the maximum width of the tidal pool. Using indirect measurement, she begins by marking the points A, C, and E. Shira measures the distances shown on the image. Next, Shira marks point B along AC and point D along AE, such that BD is parallel to CE. B A 7 ft D 3 ft 31.5 ft C E What is the distance across the tidal pool at its widest point? Chapter Skills Practice 511
48 Lesson.6 Skills Practice page Keisha is visiting a museum. She wants to know the height of one of the sculptures. She places a small mirror on the ground between herself and the sculpture, then she backs up until she can see the top of the sculpture in the mirror. x 5.5 ft 19.2 ft 13.2 ft What is the height of the sculpture? 12. Micah wants to know the height of his school. He places a small mirror on the ground between himself and the school, then he backs up until he can see the highest point of the school in the mirror. x 6 ft 93.5 ft ft What is the height of Micah s school? 512 Chapter Skills Practice
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