UNDERSTAND PRACTICE 10. Error Analysis Kendall was asked to find 16. Trace ABCD and point P. the scale factor for the dilation. What is Draw the dilation of Kendall's error? ABCD using P as the center of dilation and B p . B 6 sides that are two times 3 B as long. SEE EXAMPLE 1 4 N 4 4 D' 17. How are the side lengths of the preimage and dilated image related? 6 = 3 SEE EXAMPLE 2 O R The scale factor is 3. X 18. What is the scale factor of the dilation shown? SEE EXAMPLE 3 11. Higher Order Thinking Points M(a, b) and N(c, d) are dilated by scale factor k, with the origin as the center of dilation. Show algebraically that MN || M'N'. 12. Communicate Precisely Suppose you want to dilate a figure on the coordinate plane with a center of dilation at point (a, b) that is not the origin and with a scale factor k. Describe how 19. What are the coordinates of D1.5(ABCD) for you can use a composition of translations and A (2, 0), B(8, -4), C(4, -6), and D(-5, -10)? a dilation centered at the origin to dilate the SEE EXAMPLE 4 figure. Then write the transformation rule. 20. What are the coordinates of D(2, x)(AXYZ) for 13. Reason Rectangle J'K'L'M' is a dilation of JKLM X(1, 1), Y(2, 2), and Z(3, 0)? SEE EXAMPLE 5 with scale factor k. What are the perimeter and 21. The population density of the city of area of JKLM? Wellington is 12,000 people per square mile. K The neighboring city Morrison is a dilation of Wellington with scale factor 1.2. If Morrison has & of the population of Wellington, what is the population density of Morrison? SEE EXAMPLE 6 14. Mathematical Connections Carolina says that 22. What are the coordinates of the center of when a figure is dilated using a scale factor of dilation for the dilation shown? 2, the angle measures in the image are twice the angle measures in the preimage. How could you use the Triangle Angle Sum Theorem to explain why this cannot be true? 4 F 15. Generalize Is it always true that 2 (Dm . Dn)(X) = Dmn(X)? Explain. D G D' O 2 4 6 10 - 2 G'APPLY C Mixed Review Available Online 3. Reason The images on Henry's digital camera ASSESSMENT PRACTICE have a width-to-length ratio of 2 : 3. He wants to make an 8 in.-by-10 in. print of one of 26. Copy and complete the table to show his photographs. information about dilations centered at the origin. a. Is this possible? Explain. Preimage Scale b. How can Henry crop an image so that an Coordinates Image Factor Coordinates 8 in.-by-10 in. print can be made? (5, -2) (9 , 3 ) 24. Model With Mathematics Alex draws the scale (3, 1) 1.5 model shown as a plan for a large wall mosaic. (-6, 0) (-1, 2) (-5, 10) 27. SAT/ACT A dilation maps AABC to AA'B'C'. The area of AABC is 13 square units, and the area of AA'B'C' is 52 square units. What is the scale factor? 10 cm A 2 4 B 13 D 26 28. Performance Task Alberto wants to make a scale model of the Wright brothers' glider. 12 cm a. The wall is 10 m wide and 7 m high. What are the dimensions of the largest mosaic he Wingspan: 9.8 m Wing Area: can make on that wall? Explain. 28.3 m2 b. He will use 2-cm square tiles to make his mosaic. How many tiles will he need? Explain how you found your answer. Height: 2.4 m Length: 4.9 m 25. Look for Relationships How far from the screen should the light be placed in order for the shadow of the puppet to be 30 in. tall? Explain how you found your answer. Part A The wingspan of the scale model must be between 15 cm and 18 cm. What scale factor should he use? Explain. Part B Use your scale factor from Part A. What 30 in. will be the length, wingspan, and height of the 25 in. model glider? Part C What will be the wing area of the model glider? If both wing sections are the 2 ft same size, what will be the dimensions of each wing section?UNDERSTAND PRACTICE 9. Construct Arguments Is it possible to use only translations and dilations to map one circle to What are the vertices of each image? SEE EXAMPLE 1 another? Explain. 15. (7(5, -4) . D1.5) (AXYZ) for X(6, -2), Y(4, 1), Z( -2, 3) 10. Error Analysis Keegan was asked to graph (90. . D2)(ABC). Explain Keegan's error. 16. (Rx-axis . Do.5)(LMNP) for L(2, 4), M(4, 4), N( 4, -4), P ( 2, -4) TY 17. (Ry-axis . D2 . 270.) (APQR) for P(1, 3), Q(-4, 2), R ( 0 , 5 ) 6 18. (Do.25 . Rx-axis)(ABCD) for A(2, 6), B(0, 0), 4 C(-5, 8), D(-2, 10) B X N For each black pre-image and red image describe B' A'A C X the similarity transformation, and write a similarity -6 -4 -2 0 2 4 statement. SEE EXAMPLES 2 AND 3 19. D Q R 4 11. Mathematical Connections In the diagram, ABCD ~ A'B'C'D'. What are the angle measures P S of A'B'C'D'? - 4 -2 O N C 20. 48 D 80 6 B P 1250 A 2 X C X 12. Construct Arguments Are all squares similar? -10 -8 -6 -4 -2 0 2 4 Use transformations to explain. 13. Generalize Show whether a composition of a 21. dilation and a translation can be performed ther order and result in the same image. (Hint: Test whether the equation is true . ) ( D K o T ( 2 , by ) ( x , y ) = ( T ( a , b ) . D k ) (x , y ) 14. Higher Order Thinking Given isosceles triangle Do the figures appear to be similar? Explain. ABC with AB = BC. Point D is the midpoint SEE EXAMPLE 4 23. of AB , E is the midpoint of BC , and F is the 22. midpoint of CA. Use a similarity transformation and triangle congruence to show that AABC ~ AFEC . LESSON 7-2 Similarity Transformations 315ew Available APPLY C ASSESSMENT PRACTICE 24. Reason Can Ahmed use the larger sheets of 27. Graph the image of (T(1, -3) . D2) (AABC). paper shown to make paper cutouts similar to Pr his original hummingbird cutout? If not, how can he trim the sheets of paper so he can use B(- 1, 3) 2 C(3, 2) them? Justify your answer. X - 2 0 2 6 9 in. -2 A(5, - 2) I C 4 ft -4 rigid 12 in. triar 3 ft 28. SAT/ACT What are the coordinates of (D4 . Rx-axis) (8, 2)? 120 cm A (-32, 8) 180 cm B (32, 8) 25. Make Sense and Persevere Rachel makes a ES sketch for a stage set design on a grid. She @ (-32, -8) plans to have a gauze fabric called a scrim D (32, -8) drop down from a beam that is 5.5 m wide. Assuming that her sketch is similar to the actual 29. Performance Task Lourdes makes sketches for set, how much scrim is needed? Explain. her graphic novel using a repeating similar shape as a motif. 5.5 m Area of scrim 26. Look for Relationships Juanita wants to make a dollhouse following the pattern shown Part A On a separate sheet of paper, draw a but with a reduced size so that the floor has small simple figure. Label it A. an area of 25 in.2. Make a sketch showing the dimensions of the pieces for the smaller Part B Use transformations, including similarity dollhouse transformations, to create at least five images that are similar to figure A. Label the images B, Dollhouse Pattern C, D, E, and F. 6 in. Part C Is it possible to select any two of your RE Thi 13.5 in. 9 in. side 13.5 in. figures and find a similarity transformation that for in. 6 in. back root maps one to the other? Explain. con the In. onbumpnos elpright bi 13.5 in. 9 in. front 6 in. | floor in side in. 6 in. 9 in. wasRealize. comScan for Multimedia QPractice Tutorial PRACTICE & PROBLEM SOLVING Additional Exercises Available Online PRACTICE & PROBLEM SOLVING Q Practice Tutor PRACTICE Mixed Review Available Online UNDERSTAND APPLY 10. Construct Arguments Write a proof of the For Exercise 16-18, explain whether each pair of ASSESSMENT PRACTICE triangles is similar. SEE EXAMPLES 1-3 22. Communicate Precisely A building manager 25. Which condition is 10 Angle-Angle Similarity Theorem. needs to order 9 replacement panes that are 16. 11.25 all the same size, each similar to the window sufficient to show that Given: LT = LX itself. At what angles should each pane be cut AABC ~ AQPR? Select LU = LY in order to fit in the window? What are the all that apply 810 in Prove : ATUV ~ AXYZ dimensions of each pane? Explain. A RP = 4.5 B mLQ = 63 3.8 MLP = 81 D MAR = 81 11. Use Structure For each in 40.5 it 96 26. SAT/ACT For which value of FJ must AFGJ be triangle, name the triangle similar to AABC and explain similar to AFHG? 1 60 why it is similar. 42 in - 60 ft - 0o 50> A 6 B 8 D 12 1080 23. Use Structure The 0.0026m screen of a surveying 27. Performance Task A rescue helicopter hovering device is 0.0026 m at an altitude of 3.5 km sights a campsite just 0.1 m\\ 12. Construct Arguments If two triangles are wide and is 0.1 m over the peak of a mountain. Lens congruent by ASA, are the triangles similar? away from the lens. Explain. If the surveyor wants 19. What is FG? SEE EXAMPLES 4 AND 5 the image of the 2-m Radio Range 13. Error Analysis What is Russel's error? target to fit on the screen, what distance 3.5 km d should the lens 430 >H 180 - 80 - 60 = 40 620 be from the target? so the unlabeled angle in Explain. - 2 m - - 2.4 km - each triangle is 40. So, 430 M mLM = 60, and thus 20. What is the value of x? SEE EXAMPLES 4 AND 5 24. Mathematical Connections If a light beam Part A The horizontal distance of the AFGHANLM by AA ~ . X strikes the inside of a fiber optic cable, it helicopter from the mountain is 2.4 km. If the bounces off at the same angle. In a cable height of the mountain is 2.8 km, what is the 670 1,200 micrometers (um) long, if the beam horizontal distance d of the helicopter from 14. Construct Arguments Write a proof of the strikes the wall after 720 um what distance the campsite? Explain. Side-Angle-Side Similarity Theorem. x + y does the beam travel? Explain. Part B The groundspeed (horizontal speed) Given: LM _LN QR QS 670 of the helicopter is 1.6 km/min. When will the LL = LO helicopter reach the campsite? Explain. S 20 720 um Prove: ALMN ~ AQRS 21. Write a proof of the Side-Side-Side Similarity Part C The radio at the campsite can only Theorem . 300 um+ y transmit to a distance of 5 km. If the helicopter begins immediately to descend toward the Given: = FG = AC campsite (along the diagonal line), how far will 15. Higher Order Thinking Explain why there is no 1,200 um the pilot be, horizontally, when he contacts Side-Side-Angle Similarity Theorem. Prove: AABC ~ AEFG the campsite? LESSON 7-3 Proving Triangles Similar 323 322 TOPIC 7 Similarity Online | Savvas Realize canScan for Multimedia PRACTICE & PROBLEM SOLVING Practice PRACTICE Additional Exercises Available on PRACTICE & PROBLEM SOLVING @ Practice Tutorial UNDERSTAND APPLY Mixed Review Available Online 11. Mathematical Connections Consider AXYZ 16. In the figure, what two smaller triangles are with altitude to the hypotenuse ZW. similar to AABC? Explain. SEE EXAMPLE 1 ASSESSMENT PRACTICE 22. Reason Jake wants the profile of a hotel he is planning to be a right triangle with the 25. For each figure, write an equation that you dimensions shown. The city prohibits structures could use to find the value of x. over 100 ft at the location where he would a. like to build. Can the hotel be located there ? -21- Explain. 18 a. Describe a sequence of transformations that 17. What are the values of h and x in the right maps AXYZ to AXZW. triangle? Explain. SEE EXAMPLE 2 144 b. Describe a sequence of transformations that 160 ft 26. SAT/ACT Which triangle is similar to AABC? 120 ft maps AXYZ to AZYW. 12. Error Analysis Amaya was asked to find DC. What is Amaya's error? 18. What is the value of y in the right triangle? 23. Look for Relationships Kiyo is repairing a AACBA ACDB AABC ~ AACD by Theorem 7-4. Explain. SEE EXAMPLE 3 wooden climbing tower. B AABD DABDC BC = DC - 10 = DC 12 27. Performance Task To estimate the height 4 7.5 C 7.5 X DC = 7.5 X 10, 127.5 cm X SO DC = 10 . 68 cm - of a tree, Tia and Felix walk away from the tree until the angle of sight with the top 19. What are the values of a and b in each right and bottom of the tree is a right angle. Let h represent the height of a person's eyes and 13. Make Sense and Persevere Is CD the geometric triangle? Explain. SEE EXAMPLES 4 AND 6 68 cm - d represent the distance away from the tree. mean of AD and BD? Explain. - 127.5 cm - 25 - 625 a. He needs to cut two crossbars. What should 75 the lengths of the two crossbars be? Explain. b. Kiyo will make a notch in each crossbar in 20. What are the values of m and n in each right order to fit them together. Where should he make the notch on each crossbar ? 14. Construct Arguments Write proofs of triangle? Explain. SEE EXAMPLE 5 Explain . Theorem 7-4 and its corollaries. a. m - 4 2m a. Given: m_JLK = 90 and LM 1 JK 24. Higher Order Thinking Write a proof for Theorem 2 - 14 . Prove: AJKL ~ AJLM ~ ALKM Part A If the height of Tia's eyes is 1.6 m and Given: Right AWXY with altitude XZ to her distance away from the tree is 2.5 m, what b. Given: AJLM ~ ALKM hypotenuse WY is the height of the tree? Round to the nearest Prove : The product of the slopes of hundredth of a meter. Prove: I'M = LM M/ J. perpendicular lines is - 1. Part B If the height of Felix's eyes is 1.7 m, c. Given: AJKL ~ AILM ~ ALKM about how far from the tree is Felix if his angle of sight is a right angle? Round to the nearest Prove: 1 = /L and JK = LK 21 . What is the value of w in the right triangle ? hundredth of a meter. 15. Higher Order Thinking Suppose the altitude to Part C Suppose Tia and Felix stand the same Explain. the hypotenuse of a right triangle also bisects distance away from another tree and their the hypotenuse. What type of right triangle 10 angles of sight are right angles, what is the is it? Use the similarity of right triangles to height of the tree? Explain. explain your answer. LESSON 7-4 Similarity in Right Triangles 331Multimedia Practice PRACTICE & PROBLEM SOLVING Additional Exercises Available out PRACTICE PRACTICE & PROBLEM SOLVING @ Practice Tutorial UNDERSTAND For Exercises 17-19, find each value. Mixed Review Available Online ASSESSMENT PRACTICE 12. Error Analysis What is Benson's error? SEE EXAMPLES 1 AND 2 26. Use Structure A building in the shape of a pyramid needs to have supports repaired, and 29. What is the value of x? two parallel sections need to be reinforced. The 5_ x 10 face of the building is an equilateral triangle. What are the lengths of KO and LN? 10x = 35 x = 3.5 X 6 m 30. SAT/ACT What is the measure of side CB? m 13. Mathematical Connections What percent of 17. X 18. y 19.2 18 m the area of APQR is the area of AQRS? Explain. 20. What is the value of x? SEE EXAMPLE 3 A 4.57 B 6.4 8.96 D 9.4 31. Performance Task Emma is determining 14. Construct Arguments Write a proof of the 27. Higher Order Thinking Use the figure to prove measurements needed to simulate the distances Side-Splitter Theorem. Theorem 2-13: Two non-vertical lines are parallel in a shuffleboard computer game that she is For Exercises 21-23, find each value of x for the if and only if they have the same slope. programming. Given: MN | AC given value of y. Round to the nearest tenth. . AM _ CN SEE EXAMPLES 4 AND 5 ve: MB 15. Higher Order Thinking Suppose O, P, and Q are midpoints of the sides of ALMN. Show that ALOQ, AOMP, AQPN, and APQO are 23. y = 18 a. Assume the slopes of lines m and n are equal. congruent to each other. 21. y = 16 22 . y = 20 Use proportions in AACE and ABCD to show Part A The horizontal lines must be parallel O 16. Construct Arguments Write a 24. Write a proof of the Triangle Midsegment that min . and in proportion so that each zone of the proof for the Triangle-Angle- Theorem. b. Now assume that m In. Show that the slopes shuffleboard appears to be the same length. Bisector Theorem. Given: DG = GE, FH = HE of m and n are equal. What are the lengths w, x, and y? Given: AD bisects LA. Part B What is the length of each Prove : GH |1 DF, GH = 1DF 28. Use Structure Aisha is building a roof and horizontal segment? e: AB = DB needs to determine the lengths of CG and CF from the design shown. How can she determine Part C Which horizontal segment is closest to Use the following outline. CG and CF? What are CG and CF? the midsegment of the triangle that extends 25. Write a proof of the Corollary to the off of the screen? How do you know? . Extend CA and draw a line through Side-Splitter Theorem. D point B parallel to AD 10 ft that intersects CA at Given: & | m || n point E. AB _ DE Prove: RC G 9 ft F Show that AF = DB 33 ft Hint: Draw AF. . Then show that AAEB Label the is isosceles. intersection of AF and BE point G. LESSON 7-5 Proportions in Triangles 339 338 TOPIC 7 Similarity Go Online | Savvas Realize Cos