Geometry Unit 3 Practice

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1 Lesson Find the image of each point after the transformation (x, y) 2 x y 3, 3. 2 a. (6, 6) b. (12, 20) Geometry Unit 3 ractice 3. Triangle X(1, 6), Y(, 22), Z(2, 21) is mapped onto XʹYʹZʹ by a dilation with center (1, 22) and a scale factor of 3. Which function represents this dilation?. (x, y) (x 1 3(x 1 1), y 1 3(y 1 2)). (x, y) (x 1 3(x 2 1), y 1 3(y 2 2)). (x, y) (x 1 3(x 2 1), y 1 3(y 1 2)). (x, y) (x 1 3, y 2 6) c. (23, 10) 4. Make use of structure. In the diagram shown, a dilation maps each point (x, y) of the preimage QR to (1 1 2(x 2 1), (y 1 2)). d. 1 3, 21 (23, 4) y e. (0.1, 1.0) 2. ttend to precision. Quadrilateral (23, 3), (4, ), (6, 0), (24, 24) is mapped onto ʹʹʹʹ by a dilation in which 2. 3 The center of dilation is (0, 0). 2 x R (2, 24) Q(24, 23) 2 a. What is the scale factor? y (23, 3) (4, ) b. Is the dilation an enlargement or a reduction? 2 x (6, 0) (24, 24) 2 c. What is the center of dilation? a. What is the scale factor for this dilation? b. What are the coordinates of ʹʹʹʹ? d. What are the coordinates of the vertices of ʹQʹRʹ? 201 ollege oard. ll rights reserved. 1 Springoard Geometry, Unit 3 ractice

2 . What are the coordinates of the image of rectangle after the figure undergoes a dilation with a scale factor of 0.7 centered at the origin? y 10 (24, 10) (4, 10) 8. The sequence of similarity transformations below is applied to QR to get STV. (x, y) (2x, 2y) (2x, 3 2 2y) What are the coordinates of the vertices of STV? y (23, 2) Q (1, 2) x 2 x (24, 22) (4, 22) 2 R(1, 23) 2 Lesson What single dilation produces the same image as the composition ( o ( o ))? o, 3,40,4 4. o 3, figure is transformed by the dilation 1 o, and 4 then by another dilation. The composite of the two dilations is the same as the single dilation o, 3. What is the second dilation?. o, 33. o, 90. o, Model with mathematics. Write the sequence of similarity transformations that maps to XYZ. (26, 9) y onstruct viable arguments. Is a dilation a rigid transformation? Explain your answer. X (, 3) Z (4, 22) 10 Y(, 22) x (26, 26 ) 2 (23, 26 ) ollege oard. ll rights reserved. Springoard Geometry, Unit 3 ractice

3 Lesson Reason quantitatively. The two rectangles shown are similar. What is the value of x? onsider the two pairs of similar figures shown. 20 x Figure is similar to figure WXYZ. Which proportion NNOT be used to find side lengths in the two figures?. WX ZW... WX YZ XY WZ YZ WX onstruct viable arguments. In isosceles,. ʹʹʹ is a dilation of by a scale factor of y z 6 a. Is ʹʹʹ isosceles? Explain. Find the values of y and z. 13. The vertices of a triangle are (21, 4), (4, ), and (6, 22). fter a dilation with center (0, 0) and a scale factor of 3, the vertices are translated T(22, ). a. Is the image congruent to the original triangle? Is the image similar to the original triangle? b. What is the ratio of the perimeter of to ʹʹʹ? Lesson Reason quantitatively. onsider the triangles shown. re any of the triangles similar? Explain. b. What are the vertices of ʹʹʹ, the final image? I III II IV ollege oard. ll rights reserved. Springoard Geometry, Unit 3 ractice

4 17. Use triangles and XYZ onstruct viable arguments. Using the diagram shown, what information is needed to conclude that QR ST using each criterion? T 18 S X R Z Q a. similarity Y a. Show that and XYZ are similar by the SS similarity criterion. b. SS similarity b. Find YZ. Explain your steps. 20. In the diagram shown, E and m 3. c. Using the definition of similarity, explain why the two triangles are similar d. omplete this statement:. a. If m (x 1 3) and m (1x 1 2), find m and m E. E 18. Which of the following is NOT an abbreviation for a statement that can be used to conclude that two triangles are similar?. similarity. SS similarity. SS similarity. S similarity b. If 20, find E to the nearest tenth ollege oard. ll rights reserved. Springoard Geometry, Unit 3 ractice

5 Lesson Reason quantitatively. In the diagram shown, RS XY, m X 2, XY 30, RS 26, and ZY 20. X R 24. ersevere in solving problems. The length of the sides of Triangle I are 6 units, 10 units, and 8 units. One side of Triangle II has a length of 12 units. a. Find the lengths of the other two sides of Triangle II if the sides with lengths 6 units and 12 units are corresponding sides. a. What is m SRZ? Z S Y b. Find the other two lengths in Triangle II if the sides with lengths 10 units and 12 units are corresponding sides. b. omplete this statement: ZYX. c. What is the value of the ratio XZ expressed as a RZ reduced fraction? Expressed as a decimal to the nearest hundredth? c. Find the other two lengths in Triangle II if the sides with lengths 8 units and 12 units are corresponding sides. d. Triangle III is similar to the other two triangles, and its longest side has a length of 3 units. What are the lengths of the other two sides of Triangle III? d. If ZY 20, what is SY to the nearest hundredth? e. If XR 2, find RZ and XZ. 2. In the diagram shown,,, and. Therefore, the two small triangles are similar to each other, and to the large triangle, by similarity. Which proportion is NOT true? 22. Find E in the diagram shown. Write your answer as a decimal rounded to the nearest tenth E Triangle I has side lengths 8, 9, and 6 units. Triangle II has side lengths 8 units, 10 2 units, and 3 12 units. Show that the two triangles are similar. F ollege oard. ll rights reserved. Springoard Geometry, Unit 3 ractice

6 Lesson In the diagram shown, E FG. Which proportion is NOT true? F 29. Reason quantitatively. In the diagram shown, XY ZW QR. Use the given measurements to find each length. 10 H E G 30 X 10. Y 14. HF FG H E. HE H HG HF. E H FG F. F GE H EH a. X Z Q 36 W 16 R 27. Make use of structure. Use the diagram shown to write a proportion to illustrate the Triangle roportionality Theorem. Y X a V d W c 28. In the diagram shown, E. b Z b. XQ c. XY d. ZW 30. In the diagram shown, 7, 8, E, and m m 7m E E 2m 62 x 7 8 a. omplete this statement using segments from the diagram: b. Find and, each to the nearest tenth. a. If E, what is? b. If E, what is the value of x? ollege oard. ll rights reserved. Springoard Geometry, Unit 3 ractice

7 Lesson In the diagram shown, S T and M ST. 33. ttend to precision. In the diagram shown, S T, M ST, S 13, and SM. 13 S M T S M T a. Name two angles that are congruent to SM. b. Name an angle that is congruent to SM. c. Name an angle that is congruent to S. d. How many right triangles appear in the diagram? a. Find M. b. Find MT. c. Find ST. d. Find T. 32. Make sense of problems. This diagram shows a rectangle JG and its two diagonals. oints, F, H, and are midpoints of the sides. 34. In EF, E F and G is a segment from vertex. Which of the following statements is NOT true? E E F G G H J F a. Which segment is an altitude of E? b. Which segment is an altitude of GJ? c. Which segment is an altitude of EJ? d. Which triangles have G as an altitude?. If G is a median in isosceles triangle EF, then G forms two right triangles.. If G is an angle bisector in isosceles triangle EF, then G forms two right triangles.. If G is a point on EF, then G forms two right triangles.. If G is a perpendicular bisector of EF, then G forms two right triangles ollege oard. ll rights reserved. Springoard Geometry, Unit 3 ractice

8 3. Write a similarity statement comparing the three triangles in the diagram shown. c. Find m 1 n if t 4 and n 4. M J d. Find t if m 4 and m 1 n 13. L Lesson Use the segments in the diagram shown. Which proportion does NOT represent orollary 1 or orollary 2 of the Right Triangle ltitude Theorem? K 38. Use the diagram shown. If necessary, round your answers to the nearest tenth. s p a b r c d e q a. Find q if r 4 and s 12.. a 1 b c c a. a d d b c d. d e 1. a b e e b b. Find r if q 10 and r 1 s 0. c. Find s if q 8 and r 3. d. Find p if r s 6 and q etermine the positive geometric mean of each pair of values. Simplify any radicals. 37. Express regularity in repeated reasoning. Use the diagram shown. If necessary, round your answers to the nearest tenth. a. 2 and 100 b. 27 and 36 t n c. 1 and 4 m d. 100 and 200 a. Find t if m 2 and n 8. e. a and b b. Find m if t and n ollege oard. ll rights reserved. Springoard Geometry, Unit 3 ractice

9 40. ersevere in solving problems. In the diagram shown, and. a f 4 16 d e c a. Use a corollary of the Right Triangle ltitude Theorem to find f. b Lesson Which three numbers do NOT form a ythagorean triple?., 12, 13. 6, 8, 10. 6, 10, 14. 8, 1, supporting wire is attached to a tree at a height of 4 feet from the ground. If the length of the wire is 1 feet, what is the distance between the base of the tree and the foot of the wire? wire b. Use the base and altitude to find the area of. Show your work. 1ft 4ft? c. Use a corollary of the Right Triangle ltitude Theorem to find a and b. 43. ttend to precision. The length of a rectangular rug is 18 feet and the length of its diagonal is 22 feet. 18ft 22ft d. Using a and b as the base and height of, find the area of. Show your work. e. What do you notice about your answers to arts b and d? a. What is the width of the rug, to the nearest foot? b. What is the perimeter of the rug, to the nearest foot? c. What is the area of the rug, to the nearest square foot? d. second rectangular rug has side lengths that are the same as the width and diagonal of the original rug. What is the length of the diagonal of the second rug to the nearest foot? ollege oard. ll rights reserved. Springoard Geometry, Unit 3 ractice

10 44. Reason quantitatively. n isosceles triangle has side lengths 1 units, 1 units, and 8 units. 1 h 1 Lesson Tell whether the three lengths are the sides of an acute triangle, a right triangle, or an obtuse triangle. a. 8, 11, 12 a b. 24, 4, 1 8 a. What is the length of the altitude h from the vertex angle, to the nearest tenth? c. 3 10, 2, 1 2 b. What is the area of the triangle, to the nearest tenth? d. 12, 14, 20 c. What is the length of the altitude a from one of the base angles? d. Find the sum of the lengths of the three altitudes of the triangle. e. 9, 11, onstruct viable arguments. Tell whether or not each triangle is a right triangle. Explain your answers. a. 4. Two sides of a right triangle have lengths 1.1 cm and 30.6 cm. a. Find the length of the third side if the given lengths represent the legs of the triangle b. b. Find the length of the third side if the given lengths represent the hypotenuse and one leg of the triangle ollege oard. ll rights reserved. Springoard Geometry, Unit 3 ractice

11 48. Reason quantitatively. Two sides of a triangle are 8.6 cm and 10. cm. a. What is an inequality that represents the length s of the shortest side of the triangle? 0. For arts a e, the last number represents the length of the hypotenuse of a right triangle and the other two numbers represent the lengths of the legs. Find each missing length. a. 10,?, 1 b. What is an inequality that represents the length l of the longest side of the triangle? b. 8,?, 10 c. 100,?, 101 d. 19,?, 2 c. What is the length of the third side if it is the leg of a right triangle? e. n,?, n 3 1 d. What is the length of the third side if it is the hypotenuse of a right triangle? 49. Four students wrote the following statements about two given positive numbers. Which statement is always true?. If I select any number between the two given numbers, the three numbers can be the sides of a right triangle.. If the two numbers are the lengths of two sides of a triangle, then the sum of the two numbers can be the length of the third side of the triangle.. If I select any number between the two given numbers, the three numbers can be the lengths of the sides of a triangle.. If the two numbers are lengths of two sides of a triangle, the length for the third side can be any one of these three choices: it can be less than the sum of the two numbers, it can be greater than the difference of the two numbers, or it can be between the two numbers. Lesson Use appropriate tools strategically. Find the length of the hypotenuse of an isosceles right triangle given the length of a leg. Write each answer as an exact value and as a decimal rounded to the nearest hundredth. a. 12 in. b. 2 cm c. 7a ft d. a b units 2. Find the length of the leg of an isosceles right triangle given the length of a hypotenuse. Write each answer as an exact value and as a decimal rounded to the nearest hundredth. a. 22 in. b. 19 cm c. a ft d. c d units ollege oard. ll rights reserved. Springoard Geometry, Unit 3 ractice

12 3. In an isosceles right triangle, the length of the hypotenuse is 8 units. Which measurement is NOT associated with the triangle?. 4 2 units. 8 2 units Find the length in each isosceles right triangle. a. Find the leg if the hypotenuse is 6 2 units. Lesson Express regularity in repeated reasoning. Find the length of the longer leg and the length of the hypotenuse given the length of the shorter leg of a triangle. a. 1 in. b. 8 3 cm c. a ft b. Find the hypotenuse if the leg is 13 2 cm. d. 3 units c. Find the hypotenuse if the leg is (1 1 3) cm. d. Find the leg if the hypotenuse is 1 2 unit. 7. Find each length for a triangle. a. the shorter leg and the longer leg if the hypotenuse is 2 cm. Make sense of problems. For an isosceles right triangle, find the length of the leg and the hypotenuse with the given criterion. a. The perimeter is ( ) units. b. The perimeter is ( ) units. b. the shorter leg and the hypotenuse if the longer leg is 12 in. c. the shorter leg and the hypotenuse if the longer leg is 10 ft c. The area is 12. square units. d. the shorter leg and the longer leg if the 2 hypotenuse is 3 units d. The area is m 2 square units ollege oard. ll rights reserved. Springoard Geometry, Unit 3 ractice

13 8. In a triangle, one of the legs has a length of 30 cm. Which of the following measurements is NOT associated with this triangle? In a triangle, find the lengths of the legs and the hypotenuse with the given criterion. a. The perimeter of the triangle is (1 1 3 ) cm. Lesson Model with mathematics. Identify the indicated side in the right triangle shown. N a. the leg that is opposite angle N b. the leg that is adjacent to angle M M T c. the leg that is adjacent to angle N b. The area is 36 3 square units. d. the leg that is opposite angle M c. The triangle is half of an equilateral triangle with sides that measure 30 units. 62. Find the indicated measures in the triangle shown. Q 8 d. The perimeter is (3a 1 a 3 ) units. R S 60. Make use of structure. Find a, b, c, and d in the diagram shown. S a. QS b. m S d 608 c. raw and label a triangle XYZ so XYZ QRS and the scale factor from QRS to XYZ is 7 : 2. Indicate the measures of all sides and angles. c a d. Explain how you found the measures of the sides and angles of XYZ. R b Q 63. In a right triangle, the hypotenuse is 1 cm and a leg is 11 cm. In a similar right triangle, the hypotenuse is 9 cm. Find the lengths of the legs of the smaller triangle. Write your answers to the nearest tenth ollege oard. ll rights reserved. Springoard Geometry, Unit 3 ractice

14 64. Right triangle EF is similar to right triangle HJK. EF is larger than HJK, and the length of HK is 7. cm. Which statement describes how to calculate the length of F?. dd 7. to the length of HK.. Subtract 7. from the length of HK.. Multiply the length of HK by 7... ivide the length of HK by Make use of structure. Find the scale factor and the unknown angle measures and side lengths for the pair of similar right triangles shown. 67. Using the triangle shown, write each ratio in simplest form. M 73 T a. tan N b. tan T 48 N 17 8 c. sin T F d. cos T e. sin N E Lesson Model with mathematics. Use the triangle shown to write a fraction for each trigonometric ratio. r Q 68. Use appropriate tools strategically. Use a calculator to find each value. Round your answers to the nearest hundredth. a. sin 7.3 b. cos 42.8 q p c. tan 89.6 R a. sin b. tan Q c. cos Q d. sin Q e. tan d. cos 90 e. tan Which trigonometric ratio can be greater than 1?. sine. cosine. tangent. all three ratios ollege oard. ll rights reserved. Springoard Geometry, Unit 3 ractice

15 70. Which statement is NOT true?. sin 4 cos 4. sin 20 1 sin 0 sin 70. cos 74.7 sin 1.3. sin 3 cos Lesson Use trigonometric ratios to find the indicated side lengths in the diagram shown. Show your work and write your answers to the nearest tenth. 73. In right triangle XYZ, m Y 37 and XY 27. Which of the following is NOT a method you can use to find XZ?. Solve sin Y y z.. Solve tan Y y x.. Solve cos Y x, and then use the ythagorean z Theorem.. Find m X, and then solve cos X y z. a M Use appropriate tools strategically. The diagonal of rectangle is 42.3 cm, and it forms an angle of 3 with the shorter side of the rectangle. a. a 688 b N 38 T 42.3cm b. b a. Find the length and width of the rectangle. Show your work. 72. Use appropriate tools strategically. Find the perimeter and area of each triangle. Show your work. a m b. Use the area of to find the length of T, the altitude from vertex in. Show your work. c. Use a trigonometric ratio in T to find T b. d. ompare your results for arts b and c. Which method do you prefer? 17.8 p q ollege oard. ll rights reserved. Springoard Geometry, Unit 3 ractice

16 7. campsite at point is 600 meters from a river. One group of campers hikes to the river on a path that forms a 68 angle with the direct route to the river, and gets to the river at point. nother group of campers hikes to the river at a path that forms a 40 angle with the direct path to the river, and gets to the river at point m 77. Model with mathematics. ramp is being designed so that wheelchairs can go up the distance, which is 2. feet. Write each answer to the nearest hundredth of a degree. 2. ft a. If is 30 feet, what is the measure of? River R a. What is the length of the path from to? b. If is 30 feet, what is the measure of? b. What is the length of the path from to? c. How far apart are points and? d. How much longer is the distance from to to than the straight-line distance from to? Lesson Use appropriate tools strategically. Use a calculator to find each angle measure. e sure your calculator is in degree mode. a. sin b. tan Find all the missing sides and angles for the triangle shown Find the missing sides and angles for the triangle shown. c. cos d. tan 21 (4.0108) F 2 E e. sin ollege oard. ll rights reserved. Springoard Geometry, Unit 3 ractice

17 80. For the right triangle shown, which statement is NOT true? 82. Write the three-part statement of the Law of Sines for the triangle shown. M x8 c b a t n b. x 90 2 tan 21 a. x tan 21 a b. x cos 21 b c. x cos 21 a c N m 83. Use appropriate tools strategically. Find each measure to the nearest tenth. U T Lesson onstruct viable arguments. omplete the steps below to derive a part of the Law of Sines. V 88 a. VW W r h q b. UV Q T p R a. In QT, write an expression for sin Q. b. In RT, write an expression for sin R. c. Solve for h in arts a and b. 84. The Law of Sines NNOT be applied to which of the following?. acute triangles. obtuse triangles. right triangles. triangles where no angle measures are known d. In art c, there are two expressions for h. Set them equal to each other. e. Starting with your equation in art d, divide each side by rq ollege oard. ll rights reserved. Springoard Geometry, Unit 3 ractice

18 8. Which measure for can be found using the Law of Sines? the altitude from vertex. m. m 87. onsider the triangles below Q R T V a. Use the Law of Sines in QR to find m Q. Show your work. b. Use the Law of Sines in STV to find m T. Show your work. S Lesson Which diagram indicates two sides and the nonincluded angle?. c. ompare your answers to arts a and b with the triangles in the illustration. What do you conclude? d. How can you find the actual measure of T in STV? onsider EF with m F 40, F 32, and E 2. Find two possible values, each to the nearest whole number, for each expression. a. m E. b. EF 89. Make sense of problems. Sketch two possibilities for WXY if m W 4, YW 10, and YX ollege oard. ll rights reserved. Springoard Geometry, Unit 3 ractice

19 90. Reason quantitatively. Use your triangles from Item 89, where m W 4, YW 10, and YX 8. a. Find the two possible values for m Y, each to the nearest tenth. 94. ttend to precision. onsider the triangle shown. Find YZ to the nearest tenth. X cm b. Find the two possible values for XW, each to the nearest tenth. 11.2cm Y Z Lesson Which one of the following is a statement of the Law of osines for RQ? 9. In an isosceles triangle, the legs are 12 cm and the base is cm. a. Find the measure of the vertex angle. r q Q b. Find the measure of the base angles. p R. p 2 q 2 1 r 2 2 2pq cos. q 2 p 2 1 r 2 1 2pr cos Q. r 2 p 2 1 q 2 2 2pq cos R. r 2 p 2 1 q 2 2 2rp cos 92. Use appropriate tools strategically. onsider MNT. Find m N to the nearest tenth of a degree. M 1.7cm 26.1cm N 14.8cm T Lesson Suppose you know the measures of three parts of a triangle. Which combination of known sides and angles is NOT sufficient to let you use the Law of Sines to find other parts of the triangle?. the length of two sides and the measure of an angle opposite one of the sides. the measures of three sides of the triangle. the measures of two angles and the length of the side between the angles. the measures of two angles and the length of a side that is not between the angles 93. Which set of known measures is NOT enough to use the Law of osines?. the three sides of the triangle. two sides and one angle of the triangle. two sides and three angles of the triangle. one side and two angles of the triangle ollege oard. ll rights reserved. Springoard Geometry, Unit 3 ractice

20 97. onsider triangle QR. 99. Make use of structure. onsider triangle HJK. H R 18.7 Q 12.3 K J a. What combinations of sides and angles are shown? b. Which law can you use to find another measurement, the Law of Sines or the Law of osines? c. What is m to the nearest tenth? d. What is m Q to the nearest tenth? e. What is m R to the nearest tenth? 98. onsider triangle EF. a. What combinations of sides and angles are shown? b. Which law can you use to find another measurement, the Law of Sines or the Law of osines? c. What is HK to the nearest tenth? d. What is m K to the nearest tenth? e. What is m H to the nearest degree? F E a. What combinations of sides and angles are shown? 100. ttend to precision. Two surveyors at points and are exactly 100 meters apart. The diagram shows the angle measures from the surveyors to their supply truck at point T. T b. Which law can you use to find another measurement, the Law of Sines or the Law of osines? c. What is E to the nearest tenth? m Which surveyor is closer to the supply truck? How much closer (to the nearest tenth)? Show your work. d. What is FE to the nearest tenth? e. What is m to the nearest tenth? ollege oard. ll rights reserved. Springoard Geometry, Unit 3 ractice

Name Class Date. Investigating a Ratio in a Right Triangle

Name Class Date. Investigating a Ratio in a Right Triangle Name lass Date Trigonometric Ratios Going Deeper Essential question: How do you find the tangent, sine, and cosine ratios for acute angles in a right triangle? In this chapter, you will be working etensively