jacobcd52/hs_math_cleaned · Datasets at Hugging Face

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EB = AF ___. FC Construction Triangle Proportionality Theorem Construct a line parallel to a side of a triangle.    Use a straightedge to draw △ABC. Label E on AB. Construct ∠E ≅ ∠B. Label the ̶̶ AC as F. intersection of   EF and ̶̶   EF ǁ BC by the Converse of the Corresponding Angles Postulate. 7- 4 Applying Properties of Similar Triangles 481 481 �� E X A M P L E 1 Finding the Length of a Segment Find CY. It is given that ̶̶ XY ǁ ̶̶ BC, so AX ___ XB = AY ___ YC by the Triangle Proportionality Theorem. = 10 _ 9 _ 4 CY 9 (CY ) = 40 CY = 40 _, or 4 4 _ 9 9 1. Find PN. Substitute 9 for AX, 4 for XB, and 10 for AY. Cross Products Prop. Divide both sides by 9. Theorem 7-4-2 Converse of the Triangle Proportionality Theorem THEOREM HYPOTHESIS CONCLUSION If a line divides two sides of a triangle proportionally, then it is parallel to the third side. AE _ EB = AF _ FC   EF ǁ ̶̶ BC You will prove Theorem 7-4-2 in Exercise 23. E X A M P L E 2 Verifying Segments are Parallel ̶̶̶ MN ǁ ̶̶ KL. = 2 Verify that = 42 _ 21 = 30 _ 15 = JN ___, NL JM _ MK JN _ NL Since JM ___ MK Triangle Proportionality Theorem. ̶̶̶ MN ǁ = 2 ̶̶ KL by the Converse of the 2. AC = 36 cm, and BC = 27 cm. ̶̶ DE ǁ Verify that ̶̶ AB. Corollary 7-4-3 Two-Transversal Proportionality THEOREM HYPOTHESIS CONCLUSION If three or more parallel lines intersect two transversals, then they divide the transversals proportionally. AC _ CE = BD _ DF You will prove Corollary 7-4-3 in Exercise 24. 482 48
2 Chapter 7 Similarity �� E X A M P L E 3 Art Application ̶̶ AK ǁ An artist used perspective to draw guidelines to help her sketch a row of parallel trees. She then checked the drawing by measuring the distances between the trees. What is LN? ̶̶̶ ̶̶ CM ǁ BL ǁ KL _ = AB _ LN BD BD = BC + CD BD = 1.4 + 2.2 = 3.6 cm 2.6 _ LN = 2.4 _ 3.6 ̶̶̶ DN Given 2-Transv. Proportionality Corollary Seg. Add. Post. Substitute 1.4 for BC and 2.2 for CD. Substitute the given values. 2.4 (LN) = 3.6 (2.6) LN = 3.9 cm Cross Products Prop. Divide both sides by 2.4. 3. Use the diagram to find LM and MN to the nearest tenth. The previous theorems and corollary lead to the following conclusion. Theorem 7-4-4 Triangle Angle Bisector Theorem THEOREM HYPOTHESIS CONCLUSION An angle bisector of a triangle divides the opposite side into two segments whose lengths are proportional to the lengths of the other two sides. (△ ∠ Bisector Thm.) BD _ DC = AB _ AC You will prove Theorem 7-4-4 in Exercise 38. E X A M P L E 4 Using the Triangle Angle Bisector Theorem Find RV and VT. RV ___ VT = SR ___ ST x + 2 _ 2x + 1 = 10 _ 14 by the △ ∠ Bisector Thm. Substitute the given values. You can check your answer by substituting the values into the proportion. = SR __ RV ___ VT ST = 10 __ 5 __ 14 7 = 5 __ 5 __ 7 7 14 (x + 2) = 10 (2x + 1) 14x + 28 = 20x + 10 18 = 6x x = 3 Cross Products Prop. Dist. Prop. Simplify. Divide both sides by 6. RV = x + 2 = 3 + 2 = 5 VT = 2x + 1 Substitute 3 for x. = 2 (3) + 1 = 7 4. Find AC and DC. 7- 4 Applying Properties of Similar Triangles 483 483 ��
�� THINK AND DISCUSS 1. ̶̶ BC. Use what you know about similarity ̶̶ XY ǁ and proportionality to state as many different proportions as possible. 2. GET ORGANIZED Copy and complete the graphic organizer. Draw a figure for each proportionality theorem or corollary and then measure it. Use your measurements to write an if-then statement about each figure. 7-4 Exercises Exercises GUIDED PRACTICE. 482. 482 Find the length of each segment. ̶̶̶ DG 1. Verify that the given segments are parallel. ̶̶ AB and ̶̶ CD 3. KEYWORD: MG7 7-4 KEYWORD: MG7 Parent ̶̶ RN 2. ̶̶ TU and ̶̶ RS 4. Travel The map shows the area around p. 483 Herald Square in Manhattan, New York, and the approximate length of several streets. If the numbered streets are parallel, what is the length of Broadway between 34th St. and 35th St. to the nearest foot? 484 484 Chapter 7 Similarity �� Find the length of each segment. p. 483 6. ̶̶ QR and ̶̶ RS ̶̶ CD and ̶̶ AD 7. PRACTICE AND PROBLEM SOLVING Find the length of each segment. ̶̶ KL 8. ̶̶ XZ 9. Independent Practice For See Exercises Example 8–9 10–11 12 13–14 1 2 3 4 TEKS TEKS TAKS TAKS Verify that the given segments are parallel. Skills Practice p. S17 Application Practice p. S34 10. ̶̶ AB and ̶̶ CD 11. ̶̶̶ MN and ̶̶ QR � �� 12. Architecture The wooden treehouse has horizontal siding that is parallel to the
base. What are LM and MN to the nearest hundredth? �� Find the length of each segment. 13. ̶̶ BC and ̶̶ CD 14. ̶̶ ST and ̶̶ TU = AC_ In the figure,    BC ǁ    DE ǁ    FG. Complete each proportion. 15. AB_ BD 17. DF _ = EG _ CE = AE_ EG = 16. _ DF 18. AF _ AB 20. AB_ AC _ AC = BF_ 19. BD_ CE = _ EG 21. The bisector of an angle of a triangle divides the opposite side of the triangle into segments that are 12 in. and 16 in. long. Another side of the triangle is 20 in. long. What are two possible lengths for the third side? 7- 4 Applying Properties of Similar Triangles 485 485 �� 22. This problem will prepare you for the Multi-Step TAKS Prep on page 502. Jaclyn is building a slide rail, the narrow, slanted beam found in skateboard parks. a. Write a proportion that Jaclyn can use to calculate the length of ̶̶ CE. b. Find CE. c. What is the overall length of the slide rail AJ? 23. Prove the Converse of the Triangle Proportionality Theorem. Given: AE _ EB Prove:   EF ǁ = AF _ FC ̶̶ BC 24. Prove the Two-Transversal Proportionality Corollary. Given:   AB ǁ   CD,   CD ǁ   EF Prove: AC _ = BD _ DF CE (Hint : Draw   BE through X.) 25. Given that   PQ ǁ   RS ǁ   TU a. Find
PR, RT, QS, and SU. b. Use your results from part b to write a proportion relating the segment lengths. Find the length of each segment. 26. ̶̶ EF 27. ̶̶ ST 28. Real Estate A developer is laying out lots along Grant Rd. whose total width is 500 ft. Given the width of each lot along Chavez St., what is the width of each of the lots along Grant Rd. to the nearest foot? 29. Critical Thinking Explain how to use a sheet of lined notebook paper to divide a segment into five congruent segments. Which theorem or corollary do you use? ̶̶ ̶̶ XY ǁ DE ǁ ̶̶ BC, ̶̶ AD 30. Given that Find EC. 31. Write About It In △ABC,  AD bisects ∠BAC. Write a proportionality statement for the triangle. What theorem supports your conclusion? 486 486 Chapter 7 Similarity �� 32. Which dimensions let you conclude that SR = 12, TR = 9 SR = 16, TR = 20 ̶̶ UV ǁ ̶̶ ST? SR = 35, TR = 28 SR = 50, TR = 48 33. In △ABC, the bisector of ∠A divides ̶̶ BC into segments with lengths 16 and 20. AC = 25. Which of these could be the length of ̶̶ AB? 20 12.8 16 18.75 34. On the map, 1st St. and 2nd St. are parallel. What is the distance from City Hall to 2nd St. along Cedar Rd.? 1.8 mi 3.2 mi 4.2 mi 5.6 mi 35. Extended Response Two segments are divided proportionally. The first segment is divided into lengths 20, 15, and x. The corresponding lengths in the second segment are 16, y, and 24. Find the value of x and y. Use these values and write six proportions. CHALLENGE AND EXTEND 36. The perimeter of △ABC is 29 m. ̶̶ AD bisects ∠A. Find AB and AC. 37. Prove that if two triangles are similar, then the ratio of their corresponding
angle bisectors is the same as the ratio of their corresponding sides. 38. Prove the Triangle Angle Bisector Theorem. ̶̶ AD bisects ∠A. Given: In △ABC, = AB _ Prove: BD _ AC DC ̶̶ BX ǁ ̶̶ AD and extend ̶̶ AC to X. Use properties Plan: Draw of parallel lines and the Converse of the Isosceles Triangle Theorem to show that Then apply the Triangle Proportionality Theorem. ̶̶ AX ≅ ̶̶ AB. 39. Construction Draw of similarity to divide ̶̶ AB any length. Use parallel lines and the properties ̶̶ AB into three congruent parts. SPIRAL REVIEW Write an algebraic expression that can be used to find the nth term of each sequence. (Previous course) 40. 5, 6, 7, 8,… 41. 3, 6, 9, 12,… 42. 1, 4, 9, 16,… 43. B is the midpoint of ̶̶ AC. A has coordinates (1, 4), and B has coordinates (3, -7). Find the coordinates of C. (Lesson 1-6) Verify that the given triangles are similar. (Lesson 7-3) 44. △ABC and △ADE 45. △JKL and △MLN 7- 4 Applying Properties of Similar Triangles 487 487 2.4 mi2.1 mi2.8 miCedar Rd.Aspen Rd.1st St.2nd St.CityHallLibraryge07se_c07l04007a�� 7-5 Using Proportional Relationships TEKS G.11.D Similarity and the geometry of shape: describe the effect on perimeter, area... when... dimensions of a figure are changed.... Also G.1.B, G.5.A, G.11.A, G.11.B Objectives Use ratios to make indirect measurements. Use scale drawings to solve problems. Vocabulary indirect measurement scale drawing scale Why learn this? Proportional relationships help you find distances that cannot be measured directly. Indirect measurement is any method that uses formulas, similar figures, and/or proportions to measure an object. The following example shows one indirect measurement technique. E X A M P L E 1 Measurement
Application Eiffel Tower replica in Paris, Texas A student wanted to find the height of a statue of a pineapple in Nambour, Australia. She measured the pineapple’s shadow and her own shadow. The student’s height is 5 ft 4 in. What is the height of the pineapple? Step 1 Convert the measurements to inches. AC = 5 ft 4 in. = (5 ⋅ 12) in. + 4 in. = 64 in. BC = 2 ft = (2 ⋅ 12) in. = 24 in. EF = 8 ft 9 in. = (8 ⋅ 12) in. + 9 in. = 105 in. Step 2 Find similar triangles. Because the sun’s rays are parallel, ∠1 ≅ ∠2. Therefore △ABC ∼ △DEF by AA ∼. Step 3 Find DF. = BC_ AC_ EF DF 64 _ = 24 _ 105 DF 24 (DF) = 64 ⋅ 105 DF = 280 Corr. sides are proportional. Substitute 64 for AC, 24 for BC, and 105 for EF. Cross Products Prop. Divide both sides by 24. The height of the pineapple is 280 in., or 23 ft 4 in. 1. A student who is 5 ft 6 in. tall measured shadows to find the height LM of a flagpole. What is LM? Whenever dimensions are given in both feet and inches, you must convert them to either feet or inches before doing any calculations. 488 488 Chapter 7 Similarity 8 ft 9 in.DFEge07se_ c07105002aaAB22 ftAB1C�� A scale drawing represents an object as smaller than or larger than its actual size. The drawing’s scale is the ratio of any length in the drawing to the corresponding actual length. For example, on a map with a scale of 1 cm : 1500 m, one centimeter on the map represents 1500 m in actual distance. E X A M P L E 2 Solving for a Dimension A proportion may compare measurements that have different units. The scale of this map of downtown Dallas is 1.5 cm : 300 m. Find the actual distance between Union Station and the Dallas Public Library. Use a ruler to measure the distance between Union Station and the Dallas Public Library. The distance is 6 cm. To find the actual distance x write a proportion comparing the map distance to the actual distance. 6 _ x = 1.
5 _ 300 1.5x = 6 (300) 1.5x = 1800 x = 1200 Cross Products Prop. Simplify. Divide both sides by 1.5. The actual distance is 1200 m, or 1.2 km. 2. Find the actual distance between City Hall and El Centro College. E X A M P L E 3 Making a Scale Drawing The Lincoln Memorial in Washington, D.C., is approximately 57 m long and 36 m wide. Make a scale drawing of the base of the building using a scale of 1 cm : 15 m. Step 1 Set up proportions to find the length ℓ and width w of the scale drawing. = 1 _ 15 15w = 36 ℓ _ = 1 _ 57 15 15ℓ = 57 w _ 36 ℓ = 3.8 m w = 2.4 cm Step 2 Use a ruler to draw a rectangle with these dimensions. 3. The rectangular central chamber of the Lincoln Memorial is 74 ft long and 60 ft wide. Make a scale drawing of the floor of the chamber using a scale of 1 in. : 20 ft. 7- 5 Using Proportional Relationships 489 489 S. LamarElmS. MarketS. HoustonS. GriffinFieldS. AustinCantonYoungWoodS. ErvayS. Akard JacksonCommerceMainCity HallDallas PublicLibraryUnion StationEl CentroCollege0300 mScale30Holt, Rinehart & WinstonGeometry © 2007ge07sec07105003a Dallas Area Street map2nd proof�� Similar Triangles Similarity, Perimeter, and Area Ratios STATEMENT RATIO △ABC ∼ △DEF = BC _ EF Perimeter ratio: Similarity ratio: AB _ = AC _ DE DF perimeter △ABC __ perimeter △DEF = 6 _ = 1 _ 4 24 Area ratio: area △ABC __ = 1 _ 2 = 12 _ 24 = ( 1 _ ) area △DEF 2 2 = 1 _ 2 The comparison of the similarity ratio and the ratio of perimeters and areas of similar triangles leads to the following theorem. Theorem 7-5-1 Proportional Perimeters and Areas Theorem If the similarity ratio of two similar figures is a __, then the ratio of their perimeters b, and the ratio of their areas is a 2 __ is a __ b 2 b, or ( a __ ) b. 2 You will prove Theorem 7-5-1 in
Exercises 44 and 45. E X A M P L E 4 Using Ratios to Find Perimeters and Areas Given that △RST ∼ △UVW, find the perimeter P and area A of △UVW. The similarity ratio of △RST to △UVW is 16 __ 20, or 4 __ 5. By the Proportional Perimeters and Areas Theorem, the ratio of the triangles’ perimeters is also 4 __ 5, and the ratio of the triangles’ areas is ( 4 __ 5 ), or 16 __ 25. 2 Perimeter 36 _ = 4 _ 5 P 4P = 5 (36) P = 45 ft Area = 16 _ 48 _ 25 A 16A = 25 ⋅ 48 A = 75 ft 2 The perimeter of △UVW is 45 ft, and the area is 75 ft 2. 4. △ABC ∼ △DEF, BC = 4 mm, and EF = 12 mm. If P = 42 mm and A = 96 mm 2 for △DEF, find the perimeter and area of △ABC. THINK AND DISCUSS 1. Explain how to find the actual distance between two cities 5.5 in. apart on a map that has a scale of 1 in. : 25 mi. 2. GET ORGANIZED Copy and complete the graphic organizer. Draw and measure two similar figures. Then write their ratios. 490 490 Chapter 7 Similarity �� 7-5 Exercises Exercises KEYWORD: MG7 7-5 KEYWORD: MG7 Parent GUIDED PRACTICE 1. Vocabulary Finding distances using similar triangles is called?. ̶̶̶̶ (indirect measurement or scale drawing ). Measurement To find the height of a dinosaur in p. 488 a museum, Amir placed a mirror on the ground 40 ft from its base. Then he stepped back 4 ft so that he could see the top of the dinosaur in the mirror. Amir’s eyes were approximately 5 ft 6 in. above the ground. What is the height of the dinosaur. 489 The scale of this blueprint of an art gallery is 1 in. : 48 ft. Find the actual lengths of the following walls. ̶̶ AB ̶̶ EF 3. 5. ̶̶ CD ̶̶ FG 4. 6. 4
89 Multi-Step A rectangular classroom is 10 m long and 4.6 m wide. Make a scale drawing of the classroom using the following scales. 7. 1 cm : 1 m 8. 1 cm : 2 m 9. 1 cm : 2. Given: rectangle MNPQ ∼ rectangle RSTU p. 490 10. Find the perimeter of rectangle RSTU. 11. Find the area of rectangle RSTU. Independent Practice For See Exercises Example 12 13–14 15–17 18–19 1 2 3 4 TEKS TEKS TAKS TAKS Skills Practice p. S17 Application Practice p. S34 PRACTICE AND PROBLEM SOLVING 12. Measurement Jenny is 5 ft 2 in. tall. To find the height of a light pole, she measured her shadow and the pole’s shadow. What is the height of the pole? Space Exploration Use the following information for Exercises 13 and 14. This is a map of the Mars Exploration Rover Opportunity’s predicted landing site on Mars. The scale is 1 cm : 9.4 km. What are the approximate measures of the actual length and width of the ellipse? 13. ̶̶ KJ 14. ̶̶ NP � � � � Multi-Step A park at the end of a city block is a right triangle with legs 150 ft and 200 ft long. Make a scale drawing of the park using the following scales. 15. 1.5 in. : 100 ft 16. 1 in. : 300 ft 17. 1 in. : 150 ft 7- 5 Using Proportional Relationships 491 491 �� Given that pentagon ABCDE ∼ pentagon FGHJK, find each of the following. 18. perimeter of pentagon FGHJK 19. area of pentagon FGHJK Estimation Use the scale on the map for Exercises 20–23. Give the approximate distance of the shortest route between each pair of sites. 20. campfire and the lake 21. lookout point and the campfire 22. cabins and the dining hall 23. lookout point and the lake Given: △ABC ∼ △DEF 24. The ratio of the perimeter of △ABC to the perimeter of △DEF is 8 __ 9. What is the similarity ratio of △ABC to △DEF? 25. The ratio of the area of △
ABC to the area of △DEF is 16 __ 25. What is the similarity ratio of △ABC to △DEF? 26. The ratio of the area of △ABC to the area of △DEF is 4 __ 81. What is the ratio of the perimeter of △ABC to the perimeter of △DEF? 27. Space Exploration The scale of this model of the space shuttle is 1 ft : 50 ft. In the actual space shuttle, the main cargo bay measures 15 ft wide by 60 ft long. What are the dimensions of the cargo bay in the model? 28. Given that △PQR ∼ △WXY, find each ratio. a. perimeter of △PQR__ perimeter of △WXY b. area of △PQR __ area of △WXY c. How does the result in part a compare with the result in part b? 29. Given that rectangle ABCD ∼ EFGH. The area of rectangle ABCD is 135 in 2. The area of rectangle EFGH is 240 in 2. If the width of rectangle ABCD is 9 in., what is the length and width of rectangle EFGH? 30. Sports An NBA basketball court is 94 ft long and 50 ft wide. Make a scale drawing of a court using a scale of 1 __ 4 in. : 10 ft. 31. This problem will prepare you for the Multi-Step TAKS Prep on page 502. A blueprint for a skateboard ramp has a scale of 1 in. : 2 ft. On the blueprint, the rectangular piece of wood that forms the ramp measures 2 in. by 3 in. a. What is the similarity ratio of the blueprint to the actual ramp? b. What is the ratio of the area of the ramp on the blueprint to its actual area? c. Find the area of the actual ramp. 492 492 Chapter 7 Similarity �� Math History In 1075 C.E., Shen Kua created a calendar for the emperor by measuring the positions of the moon and planets. He plotted exact coordinates three times a night for five years. Source: history.mcs. st-andrews.ac.uk 32. Estimation The photo shows a person who is 5
ft 1 in. tall standing by a statue in Jamestown, North Dakota. Estimate the actual height of the statue by using a ruler to measure her height and the height of the statue in the photo. 33. Math History In A.D. 1076, the mathematician Shen Kua was asked by the emperor of China to produce maps of all Chinese territories. Shen created 23 maps, each drawn with a scale of 1 cm : 900,000 cm. How many centimeters long would a 1 km road be on such a map? 34. Points X, Y, and Z are the midpoints of ̶̶ JK, ̶̶ KL, and ̶̶ LJ, respectively. What is the ratio of the area of △JKL to the area of △XYZ? 35. Critical Thinking Keisha is making two scale drawings of her school. In one drawing, she uses a scale of 1 cm : 1 m. In the other drawing, she uses a scale of 1 cm : 5 m. Which of these scales will produce a smaller drawing? Explain. 36. The ratio of the perimeter of square ABCD to the perimeter of square EFGH is 4 __ 9. Find the side lengths of each square. 37. Write About It Explain what it would mean to make a scale drawing with a scale of 1 : 1. 38. Write About It One square has twice the area of another square. Explain why it is impossible for both squares to have side lengths that are whole numbers. 39. △ABC ∼ △RST, and the area of △ABC is 24 m 2. What is the area of △RST? 16 m 2 29 m 2 36 m 2 54 m 2 40. A blueprint for a museum uses a scale of 1 __ in. : 1 ft. 4 One of the rooms on the blueprint is 3 3 __ in. long. 4 How long is the actual room? 4 ft 15 ft 45 ft 180 ft 41. The similarity ratio of two similar pentagons is 9 __. What is the ratio of the 4 perimeters of the pentagons 81 _ 16 42. Of two similar triangles, the second triangle has sides half the length of the first. Given that the area of the first triangle is 16 ft 2, find the area of the second. 32 ft 2 16 ft 2 4 ft 2 8 ft 2 7- 5 Using Proportional Relationships 493 493 �� CHALLENGE
AND EXTEND 43. Astronomy The city of Eugene, Oregon, has a scale model of the solar system nearly 6 km long. The model’s scale is 1 km : 1 billion km. a. Earth is 150,000,000 km from the Sun. How many meters apart are Earth and the Sun in the model? b. The diameter of Earth is 12,800 km. What is the diameter, in centimeters, of Earth in the model? �� 44. Given: △ABC ∼ △DEF AB + BC + AC __ DE + EF +DF Prove: = AB _ DE 45. Given: △PQR ∼ △WXY Area △PQR __ Area △WXY Prove: = PR 2 _ WY 2 46. Quadrilateral PQRS has side lengths of 6 m, 7 m, 10 m, and 12 m. The similarity ratio of quadrilateral PQRS to quadrilateral WXYZ is 1 : 2. a. Find the lengths of the sides of quadrilateral WXYZ. b. Make a table of the lengths of the sides of both figures. c. Graph the data in the table. d. Determine an equation that relates the lengths of the sides of quadrilateral PQRS to the lengths of the sides of quadrilateral WXYZ. SPIRAL REVIEW Solve each equation. Round to the nearest hundredth if necessary. (Previous course) 47. (x - 3) 2 = 49 49. 4 (x + 2) 2 - 28 = 0 48. (x + 1) 2 - 4 = 0 Show that the quadrilateral with the given vertices is a parallelogram. (Lesson 6-3) 50. A (-2, -2), B (1, 0), C (5, 0), D (2, -2) 51. J (1, 3), K (3, 5), L (6, 2), M (4, 0) 52. Given that 58x = 26y, find the ratio y : x in simplest form. (Lesson 7-1) KEYWORD: MG7 Career Q: What math classes did you take in high school? A: Algebra, Geometry, and Probability and Statistics Q: What math-related classes did you take in college? A: Trigonometry, Precalculus, Drafting
, and System Design Q: How do photogrammetrists use math? A: Photogrammetrists use aerial photographs to make detailed maps. To prepare maps, I use computers and perform a lot of scale measures to make sure the maps are accurate. Q: What are your future plans? A: My favorite part of making maps is designing scale drawings. Someday I’d like to apply these skills toward architectural work. Elaine Koch Photogrammetrist 494 494 Chapter 7 Similarity �� 7-6 Dilations and Similarity in the Coordinate Plane TEKS G.11.A Similarity and the geometry of shape: use... properties and transformations to... justify conjectures.... Objectives Apply similarity properties in the coordinate plane. Use coordinate proof to prove figures similar. Vocabulary dilation scale factor Also G.2.B, G.9.B Who uses this? Computer programmers use coordinates to enlarge or reduce images. Many photographs on the Web are in JPEG format, which is short for Joint Photographic Experts Group. When you drag a corner of a JPEG image in order to enlarge it or reduce it, the underlying program uses coordinates and similarity to change the image’s size. A dilation is a transformation that changes the size of a figure but not its shape. The preimage and the image are always similar. A scale factor describes how much the figure is enlarged or reduced. For a dilation with scale factor k, you can find the image of a point by multiplying each coordinate by k: (a, b) → (ka, kb). E X A M P L E 1 Computer Graphics Application If the scale factor of a dilation is greater than 1 (k > 1), it is an enlargement. If the scale factor is less than 1 (k < 1), it is a reduction. The figure shows the position of a JPEG photo. Draw the border of the photo after a dilation with scale factor 3 __. 2 Step 1 Multiply the vertices of the photo A (0, 0), B (0, 4), C (3, 4), and D (3, 0) by 3 __ 2. Rectangle ABCD Rectangle A'B'C'D' _ _ A (0, 0) → A' (0 ⋅ 3 ) → A' (0, 00, 4) → B' (0 ⋅ 3 ) → B' (0, 63, 4)
→ C' (3 ⋅ 3 ) → C' (4.5, 63, 0) → D' (3 ⋅ 3 ) → D' (4.5, 0), 0 ⋅ 3 2 2 Step 2 Plot points A' (0, 0), B' (0, 6), C' (4.5, 6), and D' (4.5, 0). Draw the rectangle. 1. What if…? Draw the border of the original photo after a dilation with scale factor 1 __ 2. 7- 6 Dilations and Similarity in the Coordinate Plane 495 495 �� E X A M P L E 2 Finding Coordinates of Similar Triangles Given that △AOB ∼ △COD, find the coordinates of D and the scale factor. Since △AOB ∼ △COD, AO _ = OB _ CO OD = 3 _ 2 _ 4 OD Substitute 2 for AO, 4 for CO, and 3 for OB. 2OD = 12 OD = 6 Cross Products Prop. Divide both sides by 2. D lies on the x-axis, so its y-coordinate is 0. Since OD = 6, its x-coordinate must be 6. The coordinates of D are (6, 0). (3, 0) → (3 ⋅ 2, 0 ⋅ 2) → (6, 0), so the scale factor is 2. 2. Given that △MON ∼ △POQ and coordinates P (-15, 0), M (-10, 0), and Q (0, -30), find the coordinates of N and the scale factor. E X A M P L E 3 Proving Triangles Are Similar Given: A (1, 5), B (-1, 3), C (3, 4), D (-3, 1), and E (5, 3) Prove: △ABC ∼ △ADE Step 1 Plot the points and draw the triangles. Step 2 Use the Distance Formula to find the side lengths. AB = √  (-1 - 1) 2 + (3 - 5) 2 AC = √  (3 - 1
) 2 + (4 - 5 AD = √  (-3 - 1) 2 + (1 - 5) 2 AE = √  (5 - 1) 2 + (3 - 5) 2 = √  32 = 4 √  2 = √  20 = 2 √  5 Step 3 Find the similarity ratio. AB _ AD = AC _ AE √  Since AB ___ AD by SAS ∼. = AC ___ AE and ∠A ≅ ∠A by the Reflexive Property, △ABC ∼ △ADE 3. Given: R (-2, 0), S (-3, 1), T (0, 1), U (-5, 3), and V (4, 3) Prove: △RST ∼ △RUV 496 496 Chapter 7 Similarity �� E X A M P L E 4 Using the SSS Similarity Theorem Graph the image of △ABC after a dilation with scale factor 2. Verify that △A'B'C'∼ △ABC. Step 1 Multiply each coordinate by 2 to find the coordinates of the vertices of △A'B'C '. A (2, 3) → A' (2 ⋅ 2, 3 ⋅ 2) = A' (4, 6) B (0, 1) → B' (0 ⋅ 2, 1 ⋅ 2) = B' (0, 2) C (3, 0) → C' (3 ⋅ 2, 0 ⋅ 2) = C' (6, 0) Step 2 Graph △A'B'C '. Step 3 Use the Distance Formula to find the side lengths. AB = √  (2 - 0) 2 + (3 - 1) 2 A'B' = √  (4 - 0) 2 + (6 - 2 =
√  32 = 4 √  2 BC = √  (3 - 0) 2 + (0 - 1) 2 B'C'= √  (6 - 0) 2 + (0 - 2) 2 = √  10 = √  40 = 2 √  10 AC = √  (3 - 2) 2 + (0 - 3) 2 A'C'= √  (6 - 4) 2 + (0 - 6) 2 = √  10 = √  40 = 2 √  10 Step 4 Find the similarity ratio. A'B' _ AB =, B'C'_ = BC 2 √  10 _ √  10 = 2, A'C'_ = AC 2 √  10 _ √  10 = 2 Since A'B' _ = B'C' _ BC = A'C' _ AC AB, △ABC ∼ △A'B'C'by SSS ∼. 4. Graph the image of △MNP after a dilation with scale factor 3. Verify that △M'N'P' ∼ △MNP. THINK AND DISCUSS 1. △JKL has coordinates J (0, 0), K (0, 2), and L (3, 0). Its image after a dilation has coordinates J' (0, 0), K'(0, 8), and L' (12, 0). Explain how to find the scale factor of the dilation. 2. GET ORGANIZED Copy and complete the graphic organizer. Write the definition of a dilation, a property of dilations, and an example and nonexample of a dilation. 7- 6 Dilations and Similarity in the Coordinate Plane 497 497 ��
�� 7-6 Exercises Exercises KEYWORD: MG7 7-6 KEYWORD: MG7 Parent GUIDED PRACTICE Vocabulary Apply the vocabulary from this lesson to answer each question. 1. A? is a transformation that proportionally reduces or enlarges a figure, ̶̶̶̶ such as the pupil of an eye. (dilation or scale factor) 2. A ratio that describes or determines the dimensional relationship of a figure to that which it represents, such as a map scale of 1 in. : 45 ft, is called a (dilation or scale factor)?. ̶̶̶̶. Graphic Design A designer created p. 495 this logo for a real estate agent but needs to make the logo twice as large for use on a sign. Draw the logo after a dilation with scale factor 2. Given that △AOB ∼ △COD, p. 496 find the coordinates of C and the scale factor. 5. Given that △ROS ∼ △POQ, find the coordinates of S and the scale factor. Given: A (0, 0), B (-1, 1), C (3, 2), D (-2, 2), and E (6, 4) p. 496 Prove: △ ABC ∼ △ADE 7. Given: J (-1, 0), K (-3, -4), L (3, -2), M (-4, -6), and N (5, -3) Prove: △ JKL ∼ △JMN. 497 Multi-Step Graph the image of each triangle after a dilation with the given scale factor. Then verify that the image is similar to the given triangle. 9. scale factor 3 __ 2 8. scale factor 2 498 498 Chapter 7 Similarity xy440ge07se_c07l06004a�� Independent Practice For See Exercises Example 10 11–12 13–14 15–16 1 2 3 4 TEKS TEKS TAKS TAKS Skills Practice p. S17 Application Practice p. S34 PRACTICE AND PROBLEM SOLVING 10. Advertising A promoter produced this design for a street festival. She now wants to make the design smaller
to use on postcards. Sketch the design after a dilation with scale factor 1 __ 2. 11. Given that △UOV ∼ △XOY, find the coordinates of X and the scale factor. 12. Given that △MON ∼ △KOL, find the coordinates of K and the scale factor. 13. Given: D (-1, 3), E (-3, -1), F (3, -1), G (-4, -3), and H (5, -3) Prove: △DEF ∼ △DGH 14. Given: M (0, 10), N (5, 0), P (15, 15), Q (10, -10), and R (30, 20) Prove: △MNP ∼ △MQR Multi-Step Graph the image of each triangle after a dilation with the given scale factor. Then verify that the image is similar to the given triangle. 15. J (-2, 0) and K (-1, -1), and L (-3, -2) with scale factor 3 16. M (0, 4), N (4, 2), and P (2, -2) with scale factor 1 __ 2 17. Critical Thinking Consider the transformation given by the mapping (x, y) → (2x, 4y). Is this transformation a dilation? Why or why not? 18. /////ERROR ANALYSIS///// Which solution to find the scale factor of the dilation that maps △RST to △UVW is incorrect? Explain the error. 19. Write About It A dilation maps △ABC to △A'B 'C '. How is the scale factor of the dilation related to the similarity ratio of △ABC to △A'B 'C '? Explain. 20. This problem will prepare you for the Multi-Step TAKS Prep on page 502. a. In order to build a skateboard ramp, Miles draws △JKL on a coordinate plane. One unit on the drawing represents 60 cm of actual distance. Explain how he should assign coordinates for the vertices of △JKL. b. Graph the image of △JKL after a dilation with scale factor 3. 7- 6 Dilations and Similarity in the Coordinate Plane 499 499 yxc07106005a48840��
�� 21. Which coordinates for C make △COD similar to △AOB? (0, 2.4) (0, 2.5) (0, 3) (0, 3.6) 22. A dilation with scale factor 2 maps △RST to △R'S'T'. The perimeter of △RST is 60. What is the perimeter of △R'S'T'? 30 60 120 240 23. Which triangle with vertices D, E, and F is similar to △ABC? D (1, 2), E (3, 2), F (2, 0) D (-1, -2), E (2, -2), F (1, -5) D (1, 2), E (5, 2), F (3, 0) D (-2, -2), E (0, 2), F (-1, 0) 24. Gridded Resonse ̶̶ AB with endpoints A (3, 2) and B (7, 5) is dilated by a scale factor of 3. Find the length of ̶̶̶ A'B'. CHALLENGE AND EXTEND 25. How many different triangles having ̶̶ XY as a side are similar to △MNP? 26. △XYZ ∼ △MPN. Find the coordinates of Z. 27. A rectangle has two of its sides on the x- and y-axes, a vertex at the origin, and a vertex on the line y = 2x. Prove that any two such rectangles are similar. 28. △ ABC has vertices A (0, 1), B (3, 1), and C (1, 3). △DEF has vertices D (1, -1) and E (7, -1). Find two different locations for vertex F so that △ABC ∼ △DEF. SPIRAL REVIEW Write an inequality to represent the situation. (Previous
course) 29. A weight lifter must lift at least 250 pounds. There are two 50-pound weights on a bar that weighs 5 pounds. Let w represent the additional weight that must be added to the bar. Find the length of each segment, given that (Lesson 5-2) ̶̶ HF 30. 31. ̶̶ JF ̶̶ DE ≅ ̶̶ FE. 32. ̶̶ CF △SUV ∼ △SRT. Find the length of each segment. (Lesson 7-4) 33. ̶̶ RT 34. ̶̶ V T 35. ̶̶ ST 500 500 Chapter 7 Similarity �� Direct Variation Algebra In Lesson 7-6 you learned that for two similar figures, the measure of each point was multiplied by the same scale factor. Is the relationship between the scale factor and the perimeter of the figure a direct variation? y Recall from algebra that if y varies directly as x, then y = kx, or where k is the constant of variation. __ x = k, See Skills Bank page S62 Example A rectangle has a length of 4 ft and a width of 2 ft. Find the relationship between the scale factors of similar rectangles and their corresponding perimeters. If the relationship is a direct variation, find the constant of variation. Step 1 Make a table to record data. Scale Factor k 1 _ 2 2 3 4 5 Length ℓ = k (4) ℓ = 1 _ (4) = 2 2 Width w = k (2) w = 1 _ (2) = 1 2 Perimeter P = 2ℓ + 2w 2 (2) + 2 (1) = 6 8 12 16 20 4 6 8 10 24 36 48 60 Step 2 Graph the points ( 1 _, 6), (2, 24), (3, 36), (4, 48), and (5, 60). 2 Since the points are collinear and the line that contains them includes the origin, the relationship is a direct variation. Step 3 Find the equation of direct variation. y = kx 60 = k (5) Substitute 60 for y and 5 for x. 12 = k Divide both sides by 5. y = 12 x Substitute 12 for k. Thus the constant of variation is 12. Try This TAKS Grades 9–
11 Obj. 3, 8, 10 Use the scale factors given in the above table. Find the relationship between the scale factors of similar figures and their corresponding perimeters. If the relationship is a direct variation, find the constant of variation. 1. regular hexagon with side length 6 2. triangle with side lengths 3, 6, and 7 3. square with side length 3 On Track for TAKS 501 501 �� SECTION 7B Applying Similarity Ramp It Up Many companies sell plans for build-it-yourself skateboard ramps. The figures below show a ramp and the plan for the triangular support structure at the ̶̶̶ GH, side of the ramp. In the plan, ̶̶ BC. and ̶̶ EF, ̶̶ JK are perpendicular to the base ̶̶ AB, 1. The instructions call for extra pieces of wood to ̶̶ GJ, and ̶̶ JC. Given AE = 42.2 cm, reinforce find EG, GJ, and JC to the nearest tenth. ̶̶ EG, ̶̶ AE, 2. Once the support structure is built, it is covered with a triangular piece of plywood. Find the area of the piece of wood needed to cover △ABC. A separate blueprint for the ramp uses a scale of 1 cm : 25 cm. What is the area of △ABC in the blueprint? 3. Before building the ramp, you transfer the plan to a coordinate plane. Draw △ABC on a coordinate plane so that 1 unit represents 25 cm and B is at the origin. Then draw the image of △ABC after a dilation with scale factor 3 __ 2. 502 502 Chapter 7 Similarity �� Quiz for Lessons 7-4 Through 7-6 SECTION 7B 7-4 Applying Properties of Similar Triangles Find the length of each segment. ̶̶ ST 1. ̶̶ AB and ̶̶ AC 2. 3. An artist drew a picture of railroad tracks ̶̶ EF, such that the ties What is the length of ̶̶̶ GH, and ̶̶ FH? ̶̶ JK are parallel. 7-5 Using Proportional Relationships The plan for a restaurant uses the scale of 1.5 in. : 60 ft. Find the actual length of the following walls. ̶̶ AB ̶̶ CD 4
. 6. ̶̶ BC ̶̶ EF 5. 7. 8. A student who is 5 ft 3 in. tall measured her shadow and the shadow cast by a water tower shaped like a golf ball. What is the height of the tower? 7-6 Dilations and Similarity in the Coordinate Plane 9. Given: A (-1, 2), B (-3, -2), C (3, 0), D (-2, 0), and E (1, 1) Prove: △ADE ∼ △ABC 10. Given: R (0, 0), S (-2, -1), T (0, -3), U (4, 2), and V (0, 6) Prove: △RST ∼ △RUV Graph the image of each triangle after a dilation with the given scale factor. Then verify that the image is similar to the given triangle. 11. scale factor 3 12. scale factor 1.5 Ready to Go On? 503 503 ��ge07sec07rg2001aaAB40 ft5 ft 10 in.�� For a complete list of the postulates and theorems in this chapter, see p. S82. Vocabulary cross products.............. 455 proportion................. 455 scale factor................. 495 dilation.................... 495 ratio....................... 454 similar..................... 462 extremes................... 455 scale....................... 489 similar polygons............ 462 indirect measurement....... 488 scale drawing............... 489 similarity ratio.........
.... 463 means..................... 455 Complete the sentences below with vocabulary words from the list above. 1. An equation stating that two ratios are equal is called a(n)?. ̶̶̶̶? is a transformation that changes the size of a figure but not its shape. ̶̶̶̶ 2. A(n) 3. In the proportion u _ v = x _ y, the 4. A(n)? compares two numbers by division. ̶̶̶̶? are v and x. ̶̶̶̶ 7-1 Ratio and Proportion (pp. 454–459) TEKS G.5.B, G.7.B, G.7.C, G.11.B E X A M P L E S EXERCISES ■ Write a ratio expressing the slope of ℓ. slope = rise _ run 1 - 3 = = 2 _ -4 = - 1 _ 2 Write a ratio expressing the slope of each line. 5. m 6. n 7. p ■ Solve the proportion. _ x - 3 50 2 _ = 4 (x - 3) 2 = 2 (50) 4 (x - 3) 2 4 (x - 3) = 100 2 (x - 3) = 25 x - 3 = ±5 Cross Products Prop. Simplify. Divide both sides by 4. Find the square root of both sides. x - 3 = 5 or x - 3 = -5 Rewrite as two eqns. x = 8 or x = -2 Add 3 to both sides. 504 504 Chapter 7 Similarity 8. If 84 is divided into three parts in the ratio 3 : 5 : 6, what is the sum of the smallest and the largest part? 9. The ratio of the measures of a pair of sides of a rectangle is 7 : 12. If the perimeter of the rectangle is 95, what is the length of each side? Solve each proportion. y = 9 _ _ 10. 7 3 = 9 _ x 12. x _ 4 = 3x _ 14. 12 _ 32 2x = 25 _ s 11. 10 _ 4 13 24 15. = = z - 1 _ 36 2 _ 3 (y + 1) �� 7-2 Ratios in Similar Polygons (pp. 462–467) TEKS G.5
.B, G.11.A, G.11.B E X A M P L E EXERCISES ■ Determine whether △ABC and △DEF are similar. If so, write the similarity ratio and a similarity statement. Determine whether the polygons are similar. If so, write the similarity ratio and a similarity statement. 16. rectangles JKLM and PQRS 17. △TUV and △WXY It is given that ∠A ≅ ∠D and ∠B ≅ ∠E. ∠C ≅ ∠F by the Third Angles Theorem. AB ___ = AC ___ DE DF is 2 __ 3, and △ABC ∼ △DEF. = 2 __ 3. Thus the similarity ratio = BC ___ EF 7-3 Triangle Similarity: AA, SSS, and SAS (pp. 470–477) TEKS G.5.B, G.9.B, G.11.A, G.11.B E X A M P L E ■ Given: ̶̶ AB ǁ ̶̶ CD, AB = 2CD, AC = 2CE Prove: △ABC ∼ △CDE Proof: Statements Reasons ̶̶ AB ǁ ̶̶ CD 1. 1. Given 2. ∠BAC ≅ ∠DCE 2. Corr.  Post. 3. AB = 2CD, AC = 2CE 3. Given = 2 4. AB ___ CD 5. AB ___ CD = 2, AC ___ CE = AC ___ CE 6. △ABC ∼ △CDE 4. Division Prop. 5. Trans. Prop. of = 6. SAS ∼ (Steps 2, 5) EXERCISES 18. Given: JL = 1 _ JN, JK = 1 _ 3 3 Prove: △JKL ∼ △JMN JM 19. Given: ̶̶ QR ǁ ̶̶ ST Prove: △PQR ∼ △PTS 20. Given: ̶̶ BD ǁ ̶̶ CE Prove: AB (CE) = AC (BD) (Hint: After you have proved the triangles similar, look for a proportion using AB, AC, CE, and BD, the lengths of corresponding sides.) Study Guide: Review 505 505 ��
�� 7-4 Applying Properties of Similar Triangles (pp. 481–487) TEKS G.2.A, G.3.B, G.5.B, G.9.B, G.11.A, G.11.B EXERCISES Find each length. 21. CE 22. ST Verify that the given segments are parallel. 23. ̶̶ KL and ̶̶̶ MN 24. ̶̶ AB and ̶̶ CD 25. Find SU and SV. 26. Find the length of the third side of △ABC. 27. One side of a triangle is x inches longer than another side. The ray bisecting the angle formed by these sides divides the opposite side into 3-inch and 5-inch segments. Find the perimeter of the triangle in terms of x. E X A M P L E S ■ Find PQ. ̶̶ = PR ___ QR ǁ It is given that RT Triangle Proportionality Theorem. ̶̶ ST, so PQ ___ QS by the PQ = 15 _ _ 5 6 Substitute 5 for QS, 15 for PR, and 6 for RT. 6 (PQ) = 75 Cross Products Prop. PQ = 12.5 Divide both sides by 6. ̶̶ AB ǁ ̶̶ CD. = 1.5 EC _ CA ED _ DB ■ Verify that = 6 _ 4 = 4.5 _ 3 Since EC ___ = ED ___, DB CA = 1.5 ̶̶ AB ǁ ̶̶ CD by the Converse of the Triangle Proportionality Theorem. ■ Find JL and LK. Since ̶̶ JK bisects ∠LJM, JL ___ LK = JM ___ MK by the Triangle Angle Bisector Theorem. 3x - 2 _ 2x = 12.5 _ 10 Substitute the given values. 10 (3x - 2) = 12.5 (2x) Cross Products Prop. 30x - 20 = 25x Simplify. 30x = 25x + 20 Add 20 to both sides. 5x = 20 Subtract 25x from both sides. x = 4 Divide both sides by 5. JL = 3x - 2 = 3 (4) - 2 = 10 LK = 2x = 2 (4) = 8 506 506 Chapter 7 Similarity ��
�� 7-5 Using Proportional Relationships (pp. 488–494) TEKS G.1.B, G.5.A, G.11.A, G.11.B, G.11.D E X A M P L E EXERCISES ■ Use the dimensions in the diagram to find the height h of the tower. A student who is 5 ft 5 in. tall measured his shadow and a tower’s shadow to find the height of the tower. 28. To find the height of a flagpole, Casey measured her own shadow and the flagpole’s shadow. Given that Casey’s height is 5 ft 4 in., what is the height x of the flagpole? 5 ft 5 in. = 65 in. 1 ft 3 in. = 15 in. 11 ft 3 in. = 135 in. h _ 135 = 65 _ 15 15h = 65 (135) 15h = 8775 h = 585 in. Corr. sides are proportional. Cross Products Prop. Simplify. Divide both sides by 15. The height of the tower is 48 ft 9 in. 29. Jonathan is 3 ft from a lamppost that is 12 ft high. The lamppost and its shadow form the legs of a right triangle. Jonathan is 6 ft tall and is standing parallel to the lamppost. How long is Jonathan’s shadow? 7-6 Dilations and Similarity in the Coordinate Plane (pp. 495–500) E X A M P L E EXERCISES TEKS G.2.B, G.9.B, G.11.A ■ Given: A (5, -4), B (-1, -2), C (3, 0), D (-4, -1) and E (2, 2) Prove: △ABC ∼ △ADE Proof: Plot the points and draw the triangles. 30. Given: R (1, -3), S (-1, -1), T (2, 0), U (-3, 1), and V (3, 3) Prove: △RST ∼ △RUV 31. Given: J (4, 4), K (2, 3), L (4, 2), M (-4, 0), and N (4, -
4) Prove: △JKL ∼ △JMN 32. Given that △AOB ∼ △COD, find the coordinates of B and the scale factor. Use the Distance Formula to find the side lengths. AC = 2 √  5, AE = 3 √  5 AB = 2 √  10, AD = 3 √  10 = 2 _. 3 Therefore AB _ AD = AC _ AE Since corresponding sides are proportional and ∠A ≅ ∠A by the Reflexive Property, △ABC ∼ △ADE by SAS ∼. 33. Graph the image of the triangle after a dilation with the given scale factor. Then verify that the image is similar to the given triangle. K (0, 3), L (0, 0), and M (4, 0) with scale factor 3. Study Guide: Review 507 507 �� 1. Two points on ℓ are A (-6, 4) and B (10, -6). Write a ratio expressing the slope of ℓ. 2. Alana has a photograph that is 5 in. long and 3.5 in. wide. She enlarges it so that its length is 8 in. What is the width of the enlarged photograph? Determine whether the polygons are similar. If so, write the similarity ratio and a similarity statement. 3. △ABC and △MNP 4. rectangle DEFG and rectangle HJKL 5. Given: RSTU Prove: △RWV ∼ △SWT 6. Derrick is building a skateboard ramp as shown. Given that BD = DF = FG = 3 ft, find CD and EF to the nearest tenth. Find the length of each segment. 7. ̶̶ PR ̶̶̶ YW and ̶̶̶ WZ 8. 9. To find the height of a tree, a student measured the tree’s shadow and her own shadow. If the student’s height is 5 ft 8 in., what is the height of the tree? 10. The plan for a living room uses the scale of 1.5 in. : 30 ft. Use a ruler and find the length of the actual room’s diagonal ̶̶ AB. ��
�� 11. Given: A (6, 5), B (3, 4), C (6, 3), D (-3, 2), and E (6, -1) Prove: △ABC ∼ △ADE 12. A quilter designed this patch for a quilt but needs a larger version for a different project. Draw the quilt patch after a dilation with scale factor 3 __ 2. 508 508 Chapter 7 Similarity �� FOCUS ON SAT The SAT consists of seven test sections: three verbal, three math, and one more verbal or math section not used to compute your final score. The “extra” section is used to try out questions for future tests and to compare your score to previous tests. Read each question carefully and make sure you answer the question being asked. Check that your answer makes sense in the context of the problem. If you have time, check your work. You may want to time yourself as you take this practice test. It should take you about 8 minutes to complete. 1. In the figure below, the coordinates of the vertices are A (1, 5), B (1, 1), D (10, 1), and E (10, -7). If the length of the coordinates of C? ̶̶ CE is 10, what are Note: Figure not drawn to scale. (A) (4, 1) (B) (1, 4) (C) (7, 1) (D) (1, 7) (E) (6, 1) 2. In the figure below, triangles JKL and MKN are similar, and ℓ is parallel to segment JL. What is the length of ̶̶̶ KM? Note: Figure not drawn to scale. (A) 4 (B) 8 (C) 9 (D) 13 (E) 18 3. Three siblings are to share an inheritance of $750,000 in the ratio 4 : 5 : 6. What is the amount of the greatest share? (A) $125,000 (B) $187,500 (C) $250,000 (D) $300,000 (E) $450,000 4. A 35-foot flagpole casts a 9-foot shadow at the same time that a girl casts a 1
.2-foot shadow. How tall is the girl? (A) 3 feet 8 inches (B) 4 feet 6 inches (C) 4 feet 7 inches (D) 4 feet 8 inches (E) 5 feet 6 inches 5. What polygon is similar to every other polygon of the same name? (A) Triangle (B) Parallelogram (C) Rectangle (D) Square (E) Trapezoid College Entrance Exam Practice 509 509 �� Any Question Type: Interpret A Diagram When a diagram is included as part of a test question, do not make any assumptions about the diagram. Diagrams are not always drawn to scale and can be misleading if you are not careful. Multiple Choice What is DE? 3.6 4 4.8 9 Make your own sketch of the diagram. Separate the two triangles so that you are able to find the side length measures. By redrawing the diagram, it is clear that the two triangles are similar. Set up a proportion to find DE. AB _ BC 6 _ 10 48 _ 10 DE = 4.8 = DE _ EF = DE _ 8 = DE The correct choice is C. Gridded Response △X′ Y ′Z′ is the image of △XYZ after a dilation with scale factor 1 __. Find X ′Z′. 2 Before you begin, look at the scale of both the x-axis and the y-axis. Do not assume that the scale is always 1. At first glance, you might assume that XZ is 4. But by looking closely at the x-axis, notice that each increment represents 2 units. So XZ is actually 8. When △XYZ is dilated by a factor of 1 _ 2, X′Z′ will be half of XZ. XZ = 1 _ X′Z′ = 1 _ 2 2 (8) = 4 510 510 Chapter 7 Similarity �� If the diagram does not match the given information, draw one that is more accurate. Item C Gridded Response What is the measure of MN? Read each test item and answer the questions that follow. Item A Multiple Choice Which ratio is the slope of m? 1 _ 15 1 _ 3 3 15 1. What is the scale of the y-axis? Use this scale to determine the rise of the slope. 2. What is the scale of the x-axis? Use
this scale to determine the run of the slope. 3. Write the ratio that represents the slope of m. 4. Anna selected choice B as her answer. Is she correct? If not, what do you think she did wrong? Item B Gridded Response If ABCD ∼ MNOP and AC is 6, what is AB? 5. Examine the figures. Do you think longer or shorter than ̶̶̶ MN? ̶̶ AB is 8. Describe how redrawing the figure can help you better understand the given information. 9. After reading this test question, a student redrew the figure as shown below. Explain if it is a correct interpretation of the original figure. If it is not, redraw and/or relabel it so that it is correct. Item D Multiple Choice Which is a similarity ratio for the triangles shown? 20 _ 1 10 _ 1 2 _ 1 15 _ 1 6. Do you think the drawings actually represent the given information? If not, explain why. 7. Create your own sketch of the figures to more accurately match the given information. 10. Chad determined that choice D was correct. Do you agree? If not, what do you think he did wrong? 11. Redraw the figures so that they are easier to understand. Write three statements that describe which vertices correspond to each other and three statements that describe which sides correspond to each other. TAKS Tackler 511 511 �� KEYWORD: MG7 TestPrep CUMULATIVE ASSESSMENT, CHAPTERS 1–7 Multiple Choice 1. Which similarity statement is true for rectangles ABCD and MNPQ, given that AB = 3, AD = 4, MN = 6, and NP = 4.5? Rectangle ABCD ∼ rectangle MNPQ 5. If 12x = 16y, what is the ratio of x to y in simplest form Rectangle ABCD ∼ rectangle PQMN Use the diagram for Items 6 and 7. Rectangle ABCD ∼ rectangle MPNQ Rectangle ABCD ∼ rectangle QMNP 2. △ ABC has perpendicular bisectors If AP = 6 and ZP = 4.5, what is the length of to the nearest tenth? ̶̶ XP, ̶̶ YP, and ̶̶ BC ̶̶ ZP. 6. Given that ̶̶ AB ≅ ̶̶ CD, which additional information would
be sufficient to prove that ABCD is a parallelogram? ̶̶ AB ǁ ̶̶ AC ǁ ̶̶ CD ̶̶ BD ∠CAB ≅ ∠CDB E is the midpoint of ̶̶̶ AD. 7. If   AC is parallel to   BD and m∠1 + m∠2 = 140°, what is the measure of ∠3? 20° 40° 50° 70° 8. If of ̶̶ AC is twice as long as ̶̶ DC? ̶̶ AB, what is the length 2.5 centimeters 3.75 centimeters 5 centimeters 15 centimeters 4.0 7.9 9.0 12.7 3. What is the converse of the statement “If a quadrilateral has 4 congruent sides, then it is a rhombus”? If a quadrilateral is a rhombus, then it has 4 congruent sides. If a quadrilateral does not have 4 congruent sides, then it is not a rhombus. If a quadrilateral is not a rhombus, then it does not have 4 congruent sides. If a rhombus has 4 congruent sides, then it is a quadrilateral. 4. A blueprint for a hotel uses a scale of 3 in. : 100 ft. On the blueprint, the lobby has a width of 1.5 in. and a length of 2.25 in. If the carpeting for the lobby costs $1.25 per square foot, how much will the carpeting for the entire lobby cost? $312.50 $1406.25 $3000.00 $4687.50 512 512 Chapter 7 Similarity �� When writing proportions for similar figures, make sure that each ratio compares corresponding side lengths in each figure. STANDARDIZED TEST PREP Short Response 17. △ ABC has vertices A (-2, 0), B (2, 2), and C (2, -2). △DEC has vertices D (0, -1), E (2, 0), and C (2, -2). Prove that △ ABC ∼ △DEC. 9. What type of triangle has angles that measure (2x) °, (3x - 9)
°, and (x + 27) °? 18. ∠TUV in the diagram below is an obtuse angle. Isosceles acute triangle Isosceles right triangle Scalene acute triangle Scalene obtuse triangle Use the diagram for Items 10 and 11. 10. Which of these points is the orthocenter of △FGH? F G H J Write an inequality showing the range of possible measurements for ∠TUW. Show your work or explain your answer. 19. △ ABC and △ ABD share side ̶̶ AB. Given that △ABC ∼ △ABD, use AAS to explain why these two triangles must also be congruent. 20. Rectangle ABCD has a length of 2.6 cm and a width of 1.8 cm. Rectangle WXYZ has a length of 7.8 cm and a width of 5.4 cm. Determine whether rectangle ABCD is similar to rectangle WXYZ. Explain your reasoning. 11. Which of the following could be the side lengths 21. If △ABC and △XYZ are similar triangles, there are of △FGH? FG = 2, GH = 3, and FH = 4 FG = 4, GH = 5, and FH = 6 FG = 5, GH = 4, and FH = 3 FG = 6, GH = 8, and FH = 10 12. The measure of one of the exterior angles of a right triangle is 120°. What are the measures of the acute interior angles of the triangle? 30° and 60° 40° and 50° 40° and 80° 60° and 60° Gridded Response 13. The ratio of a football field’s length to its width is 9 : 4. If the length of the field is 360 ft, what is the width of the field in feet? 14. The sum of the measures of the interior angles of a convex polygon is 1260°. How many sides does the polygon have? 15. In kite PQRS, ∠P and ∠R are opposite angles. If m∠P = 25° and m∠R = 75°, what is the measure of ∠Q in degrees? 16. Heather is 1.6 m tall and casts a shadow of 3.5 m. At the same time, a barn casts a shadow of 17.5 m. Find the height of the barn
in meters. six possible similarity statements. a. What is the probability that △ABC ∼ △XYZ is correct? b. If △ABC and △XYZ are isosceles, what is the probability that △ABC ∼ △XYZ? c. If △ABC and △XYZ are equilateral, what is the probability that △ABC ∼ △XYZ? Explain. Extended Response 22. a. Given: △SRT ∼ △VUW and ̶̶ VU ≅ ̶̶̶ VW Prove: ̶̶ SR ≅ ̶̶ ST b. Explain in words how you determine the possible values for x and y that would make the two triangles below similar. Note: Triangles not drawn to scale. c. Explain why x cannot have a value of 1 if the two triangles in the diagram above are similar. Cumulative Assessment, Chapters 1–7 513 513 �� Right Triangles and Trigonometry 8A Trigonometric Ratios 8-1 Similarity in Right Triangles Lab Explore Trigonometric Ratios 8-2 Trigonometric Ratios 8-3 Solving Right Triangles 8B Applying Trigonometric Ratios 8-4 Angles of Elevation and Depression Lab Indirect Measurement Using Trigonometry 8-5 Law of Sines and Law of Cosines 8-6 Vectors Ext Trigonometry and the Unit Circle KEYWORD: MG7 ChProj Standing over 567 feet tall, the San Jacinto Monument in LaPorte is the tallest memorial column in the world. 514 514 Chapter 8 Vocabulary Match each term on the left with a definition on the right. 1. altitude A. a comparison of two numbers by division 2. proportion 3. ratio 4. right triangle B. a segment from a vertex to the midpoint of the opposite side of a triangle C. an equation stating that two ratios are equal D. a perpendicular segment from the vertex of a triangle to a line containing the base E. a triangle that contains a right angle Identify Similar Figures Determine if the two triangles are similar. 5. 6. Special Right Triangles Find the value of x. Give the answer in simplest radical form. 7. 8. 9. 10. Solve Multi-Step Equations Solve each equation. 11. 3 (x - 1) = 12 13. 6 = 8 (x
- 3) 12. -2 (y + 5) = -1 14. 2 = -1 (z + 4) Solve Proportions Solve each proportion. 15. 4_ y = 6_ 18 16. 5_ 8 = x_ 32 17. m_ 9 = 8_ 12 18. y_ 4 = 9_ y Rounding and Estimation Round each decimal to the indicated place value. 19. 13.118; hundredth 20. 37.91; tenth 21. 15.992; tenth 22. 173.05; whole number Right Triangles and Trigonometry 515515 �� Key Vocabulary/Vocabulario angle of depression ángulo de depresión angle of elevation ángulo de elevación cosine coseno geometric mean media geométrica sine tangent seno tangente trigonometric ratio razón trigonométrica vector vector Vocabulary Connections To become familiar with some of the vocabulary terms in the chapter, consider the following. You may refer to the chapter, the glossary, or a dictionary if you like. 1. The term angle of elevation includes the word elevation. What does elevate mean in everyday usage? What do you think an angle of elevation might be? 2. A vector is sometimes defined as “a directed line segment.” How can you use this definition to understand this term? 3. The word trigonometric comes from the Greek word trigonon, which means “triangle,” and the suffix metric, which means “measurement.” Based on this, how do you think you might use a trigonometric ratio? Geometry TEKS 8-2 Tech. Lab Les. 8-1 Les. 8-2 Les. 8-3 Les. 8-4 8-4 Geo. Lab Les. 8-5 Les. 8-6 Ext. G.1.B Geometric structure* recognize the historical development of ★ geometric systems... G.2.A Geometric structure* use constructions to explore attributes of geometric figures and to make conjectures about geometric relationships ★ G.5.B Geometric patterns* use … numeric and geometric patterns to make generalizations about geometric properties, including … ratios in similar figures... ★ ★ ★ ★ ★ G.5.D Geometric patterns* identify and apply patterns from right ★ ★ ★ ★ ★ ★ triangles to solve
meaningful problems … G.7.A Dimensionality and the geometry of location* use … two- ★ ★ ★ ★ dimensional coordinate systems … G.8.C Congruence and the geometry of size* … use the ★ ★ ★ Pythagorean Theorem G.10.A Congruence and geometry of size* use congruence transformations to … justify properties of geometric figures including figures represented on a coordinate plane G.11.A Similarity and the geometry of shape* use and extend similarity properties and transformations to explore and justify conjectures about geometric figures G.11.B Similarity and the geometry of shape* use ratios to solve problems involving similar figures G.11.C Similarity and the geometry of shape* develop, apply, and justify triangle similarity relationships, such as right triangle ratios, trigonometric ratios, … ★ ★ ★ ★ ★ ★ ★ ★ ★ ★ ★ ★ ★ ★ ★ ★ ★ * Knowledge and skills are written out completely on pages TX28–TX35. 516 516 Chapter 8 Reading Strategy: Read to Understand As you read a lesson, read with a purpose. Lessons are about one or two specific objectives. These objectives are at the top of the first page of every lesson. Reading with the objectives in mind can help you understand the lesson. Identify similar polygons. Lesson 7-2 Ratios in Similar Polygons Figures that are similar (∼) have the same shape but not necessarily the same size. △1 is similar to △2 (△1 ∼ △2). △1 is not similar to △3 (△1 ≁ △3). Identify the objectives of the lesson. Read through the lesson to find where the objectives are explained. • Can two polygons be both similar and congruent? • In Example 1, the triangles are not oriented the same. How can you tell which angles are congruent and which sides are corresponding? List any questions, problems, or trouble spots you may have. • Similarity is represented by the symbol ∼. Congruence is represented by the symbol ≅. • Similar: same shape but not necessarily the same size Write down any new vocabulary or symbols. Try This Use Lesson 8-1 to complete each of the following. 1. What are the objectives of the lesson? 2. Identify any new vocabulary, formulas, and symbols. 3. Identify any examples that you need clarified. 4. Make a list of questions you
need answered during class. Right Triangles and Trigonometry 517 517 �� 8-1 Similarity in Right Triangles TEKS G.11.C Similarity and the geometry of shape: develop, apply, and justify triangle similarity relationships, such as right triangle ratios.... Objectives Use geometric mean to find segment lengths in right triangles. Apply similarity relationships in right triangles to solve problems. Vocabulary geometric mean Also G.5.B, G.5.D, G.8.A, G.8.C, G.11.A, G.11.B Why learn this? You can use similarity relationships in right triangles to find the height of Big Tex. Big Tex debuted as the official symbol of the State Fair of Texas in 1952. This 6000-pound cowboy wears size 70 boots and a 75-gallon hat. In this lesson, you will learn how to use right triangle relationships to find Big Tex’s height. In a right triangle, an altitude drawn from the vertex of the right angle to the hypotenuse forms two right triangles. Theorem 8-1-1 The altitude to the hypotenuse of a right triangle forms two triangles that are similar to each other and to the original triangle. △ABC ∼ △ACD ∼ △CBD PROOF PROOF Theorem 8-1-1 ̶̶ CD. Given: △ABC is a right triangle with altitude Prove: △ABC ∼ △ACD ∼ △CBD Proof: The right angles in △ABC, △ACD, and △CBD are all congruent. By the Reflexive Property of Congruence, ∠A ≅ ∠A. Therefore △ABC ∼ △ACD by the AA Similarity Theorem. Similarly, ∠B ≅ ∠B, so △ABC ∼ △CBD. By the Transitive Property of Similarity, △ABC ∼ △ACD ∼ △CBD. E X A M P L E 1 Identifying Similar Right Triangles Write a similarity statement comparing the three triangles. Sketch the three right triangles with the angles of the triangles in corresponding positions. By Theorem 8-1-1, △RST ∼ △SPT ∼ △RPS. 1. Write a similarity statement comparing the three triangles. 518 518 Chapter 8 Right Triangles and Trigonometry �� Consider the
proportion a __ x = x __ the same number, and that number is the geometric mean of the extremes. The geometric mean of two positive numbers is the positive square root of their product. So the geometric mean of a and b is the positive number x such that x = √  ab, or x 2 = ab.. In this case, the means of the proportion are b E X A M P L E 2 Finding Geometric Means Find the geometric mean of each pair of numbers. If necessary, give the answer in simplest radical form. A 4 and 9 Let x be the geometric mean. x 2 = (4) (9) = 36 x = 6 Def. of geometric mean Find the positive square root. B 6 and 15 Let x be the geometric mean. x 2 = (6) (15) = 90 x = √  90 = 3 √  10 Def. of geometric mean Find the positive square root. Find the geometric mean of each pair of numbers. If necessary, give the answer in simplest radical form. 2a. 2 and 8 2b. 10 and 30 2c. 8 and 9 You can use Theorem 8-1-1 to write proportions comparing the side lengths of the triangles formed by the altitude to the hypotenuse of a right triangle. All the relationships in red involve geometric means Corollaries Geometric Means COROLLARY EXAMPLE DIAGRAM 8-1-2 The length of the altitude to the hypotenuse of a right triangle is the geometric mean of the lengths of the two segments of the hypotenuse. 8-1-3 The length of a leg of a right triangle is the geometric mean of the lengths of the hypotenuse and the segment of the hypotenuse adjacent to that leg. h 2 = xy a 2 = xc b 2 = yc 8-1 Similarity in Right Triangles 519 519 �� E X A M P L E 3 Finding Side Lengths in Right Triangles Once you’ve found the unknown side lengths, you can use the Pythagorean Theorem to check your answers. Find x, y, and z. x 2 = (2) (10) = 20 x = √  20 = 2 √  5 y 2 = (12) (10) = 120 120 = 2 √  30 y = √ �
� z 2 = (12) (2) = 24 z = √  24 = 2 √  6 x is the geometric mean of 2 and 10. Find the positive square root. y is the geometric mean of 12 and 10. Find the positive square root. z is the geometric mean of 12 and 2. Find the positive square root. 3. Find u, v, and w. E X A M P L E 4 Measurement Application To estimate the height of Big Tex at the State Fair of Texas, Michael steps away from the statue until his line of sight to the top of the statue and his line of sight to the bottom of the statue form a 90° angle. His eyes are 5 ft above the ground, and he is standing 15 ft 3 in. from Big Tex. How tall is Big Tex to the nearest foot? Let x be the height of Big Tex above eye level. 15 ft 3 in. = 15.25 ft Convert 3 in. to 0.25 ft. (15.25) 2 = 5x 15.25 is the geometric mean of 5 and x. x = 46.5125 ≈ 47 Solve for x and round. Big Tex is about 47 + 5, or 52 ft tall. 4. A surveyor positions himself so that his line of sight to the top of a cliff and his line of sight to the bottom form a right angle as shown. What is the height of the cliff to the nearest foot? THINK AND DISCUSS 1. Explain how to find the geometric mean of 7 and 21. 2. GET ORGANIZED Copy and complete the graphic organizer. Label the right triangle and draw the altitude to the hypotenuse. In each box, write a proportion in which the given segment is a geometric mean. 520 520 Chapter 8 Right Triangles and Trigonometry �� 8-1 Exercises Exercises KEYWORD: MG7 8-1 KEYWORD: MG7 Parent GUIDED PRACTICE 1. Vocabulary In the proportion 2 __ 8 = 8 __ 32, which number is the geometric mean of the other two numbers Write a similarity statement comparing the three triangles in each diagram. p. 518 2. 3. 4. 519 Find the geometric mean of each pair of numbers. If
necessary, give the answer in simplest radical form. 5. 2 and 50 8. 9 and 12 6. 4 and 16 9. 16 and 25 7. 1 _ 2 and 8 10. 7 and 11 Find x, y, and z. p. 520 11. 12. 13. 520 14. Measurement To estimate the length of the USS Constitution in Boston harbor, a student locates points T and U as shown. What is RS to the nearest tenth? PRACTICE AND PROBLEM SOLVING Independent Practice Write a similarity statement comparing the three triangles in each diagram. For See Exercises Example 15. 16. 17. 15–17 18–23 24–26 27 1 2 3 4 TEKS TEKS TAKS TAKS Skills Practice p. S18 Application Practice p. S35 Find the geometric mean of each pair of numbers. If necessary, give the answer in simplest radical form. 18. 5 and 45 21. 1 _ 4 and 80 Find x, y, and z. 24. 19. 3 and 15 22. 1.5 and 12 20. 5 and 8 and 27 _ 23. 2 _ 40 3 25. 26. 8-1 Similarity in Right Triangles 521 521 �� 27. Measurement To estimate the height of the Taipei 101 tower, Andrew stands so that his lines of sight to the top and bottom of the tower form a 90° angle. What is the height of the tower to the nearest foot? 28. The geometric mean of two numbers is 8. One of the numbers is 2. Find the other number. 29. The geometric mean of two numbers is 2 √  5. One of the numbers is 6. Find the other number. Use the diagram to complete each equation. 30 33.? 31.? _ u = u _ x 34. (?) 2 = y (x + y) 32. x + y _ v 35. u 2 = (x + y) (?) = v _? Give each answer in simplest radical form. 36. AD = 12, and CD = 8. Find BD. 37. AC = 16, and CD = 5. Find BC. 38. AD = CD = √  2. Find BD. 39. BC = √  5, and AC = √  10. Find CD. 40. Finance
An investment returns 3% one year and 10% the next year. The average rate of return is the geometric mean of the two annual rates. What is the average rate of return for this investment to the nearest tenth of a percent? 41. /////ERROR ANALYSIS///// Two students were asked to find EF. Which solution is incorrect? Explain the error. 42. The altitude to the hypotenuse of a right triangle divides the hypotenuse into segments that are 2 cm long and 5 cm long. Find the length of the altitude to the nearest tenth of a centimeter. 43. Critical Thinking Use the figure to show how Corollary 8-1-3 can be used to derive the Pythagorean Theorem. (Hint: Use the corollary to write expressions for a 2 and b 2. Then add the expressions.) 44. This problem will prepare you for the Multi-Step TAKS Prep on page 542. Before installing a utility pole, a crew must first dig a hole and install the anchor for the guy wire ̶̶ RT, that supports the pole. In the diagram, ̶̶̶ ̶̶̶ WT, RS = 4 ft, and ST = 3 ft. RW ⊥ a. Find the depth of the anchor ̶̶̶ SW to the ̶̶̶ SW ⊥ nearest inch. b. Find the length of the rod ̶̶̶ RW to the nearest inch. 522 522 Chapter 8 Right Triangles and Trigonometry � � � � �� 5 ft91 ft 3 in.ge07sec08l01003a�� 45. Write About It Suppose the rectangle and square have the same area. Explain why s must be the geometric mean of a and b. 46. Write About It Explain why the geometric mean of two perfect squares must be a whole number. 47. Lee is building a skateboard ramp based on the plan shown. Which is closest to the length of the ramp from point X to point Y? 4.9 feet 5.7 feet 8.5 feet 9.4 feet 48. What is the area of △ABC? 18 square meters
36 square meters 39 square meters 78 square meters 49. Which expression represents the length of ̶̶ RS? √  y + 1) CHALLENGE AND EXTEND 50. Algebra An 8-inch-long altitude of a right triangle divides the hypotenuse into two segments. One segment is 4 times as long as the other. What are the lengths of the segments of the hypotenuse? 51. Use similarity in right triangles to find x, y, and z. 52. Prove the following. If the altitude to the hypotenuse of a right triangle bisects the hypotenuse, then the triangle is a 45°-45°-90° right triangle. 53. Multi-Step Find AC and AB to the nearest hundredth. SPIRAL REVIEW Find the x-intercept and y-intercept for each equation. (Previous course) 54. 3y + 4 = 6x 55. x + 4 = 2y 56. 3y - 15 = 15x The leg lengths of a 30°-60°-90° triangle are given. Find the length of the hypotenuse. (Lesson 5-8) 57. 3 and √  27 58. 7 and 7 √  3 59. 2 and 2 √  3 For rhombus ABCD, find each measure, given that m∠DEC = 30y°, m∠EDC = (8y + 15) °, AB = 2x + 8, and BC = 4x. (Lesson 6-4) 61. m∠EDA 60. m∠EDC 62. AB 8-1 Similarity in Right Triangles 523 523 �� 8-2 Explore Trigonometric Ratios In a right triangle, the ratio of two side lengths is known as a trigonometric ratio. Use with Lesson 8-2 Activity TEKS G.2.A Geometric structure: use constructions to explore attributes of geometric figures and to make conjectures about geometric relationships. Also G.5.B, G.9.B, G.11.A KEYWORD: MG7 Lab8 1 Construct three points and label them A, B, and C. Construct rays endpoint A. Move C so that ∠A is an acute angle. �
�� AC with common  AB and 2 Construct point D on  AC. Construct a line  AB. Label the through D perpendicular to intersection of the perpendicular line and  AB as E. 3 Measure ∠A. Measure DE, AE, and AD, the side lengths of △AED. 4 Calculate the ratios DE _ AD, AE _ AD, and DE _. AE Try This 1. Drag D along  AC. What happens to the measure of ∠A as D moves? What postulate or theorem guarantees that the different triangles formed are similar to each other? 2. As you move D, what happens to the values of the three ratios you calculated? Use the properties of similar triangles to explain this result. 3. Move C. What happens to the measure of ∠A? With a new value for m∠A, note the values of the three ratios. What happens to the ratios if you drag D? 4. Move C until DE ___ AD = AE ___ AD. What is the value of DE ___ AE? What is the measure of ∠A? Use the properties of special right triangles to justify this result. 524 524 Chapter 8 Right Triangles and Trigonometry 8-2 Trigonometric Ratios TEKS G.11.C Similarity and the geometry of shape: develop, apply, and justify triangle similarity relationships, such as... trigonometric ratios.... Objectives Find the sine, cosine, and tangent of an acute angle. Use trigonometric ratios to find side lengths in right triangles and to solve real-world problems. Vocabulary trigonometric ratio sine cosine tangent Also G.5.B, G.5.D, G.8.C, G.11.B Who uses this? Contractors use trigonometric ratios to build ramps that meet legal requirements. According to the Americans with Disabilities Act (ADA), the maximum slope allowed for a wheelchair ramp is 1__ 12, which is an angle of about 4.8°. Properties of right triangles help builders construct ramps that meet this requirement. By the AA Similarity Postulate, a right triangle with a given acute angle is similar to every other right triangle with that same acute angle measure. So △ABC ∼ △
DEF ∼△XYZ, and BC ___. These are trigonometric AC ratios. A trigonometric ratio is a ratio of two sides of a right triangle. = EF ___ DF = YZ ___ XZ Trigonometric Ratios DEFINITION The sine of an angle is the ratio of the length of the leg opposite the angle to the length of the hypotenuse. The cosine of an angle is the ratio of the length of the leg adjacent to the angle to the length of the hypotenuse. The tangent of an angle is the ratio of the length of the leg opposite the angle to the length of the leg adjacent to the angle. In trigonometry, the letter of the vertex of the angle is often used to represent the measure of that angle. For example, the sine of ∠A is written as sin A. DIAGRAM SYMBOLS __ _ a = c _ __ b = c opposite leg hypotenuse opposite leg hypotenuse sin A = sin B = cos A = cos B = _ __ b = c _ __ a = c adjacent leg hypotenuse adjacent leg hypotenuse tan A = tan B = _ __ a = b _ __ b = a opposite leg adjacent leg opposite leg adjacent leg E X A M P L E 1 Finding Trigonometric Ratios Write each trigonometric ratio as a fraction and as a decimal rounded to the nearest hundredth. A sin R sin R = 12 _ 13 ≈ 0.92 The sine of an ∠ is opp. leg _ hyp.. 8-2 Trigonometric Ratios 525 525 �� Write each trigonometric ratio as a fraction and as a decimal rounded to the nearest hundredth. B cos R cos R = 5 _ 13 ≈ 0.38 C tan S tan S = 5 _ 12 ≈ 0.42 The cosine of an ∠ is adj. leg _. hyp. The tangent of an ∠ is opp. leg _. adj. leg Write each trigonometric ratio as a fraction and as a decimal rounded to the nearest hundredth. 1a. cos A 1b. tan B 1c. sin B E X A M P L E 2 Finding Trigonometric Ratios in Special Right Triangles Use a special right triangle to write sin 60° as a fraction. Draw and label a 30°-60°-90° △. sin 60° = s √
 3 _ 2s = √  3 _ 2 The sine of an ∠ is opp. leg _ hyp.. 2. Use a special right triangle to write tan 45° as a fraction. E X A M P L E 3 Calculating Trigonometric Ratios Use your calculator to find each trigonometric ratio. Round to the nearest hundredth. A cos 76° B sin 8° C tan 82° Be sure your calculator is in degree mode, not radian mode. cos 76° ≈ 0.24 sin 8° ≈ 0.14 tan 82° ≈ 7.12 Use your calculator to find each trigonometric ratio. Round to the nearest hundredth. 3a. tan 11° 3b. sin 62° 3c. cos 30° The hypotenuse is always the longest side of a right triangle. So the denominator of a sine or cosine ratio is always greater than the numerator. Therefore the sine and cosine of an acute angle are always positive numbers less than 1. Since the tangent of an acute angle is the ratio of the lengths of the legs, it can have any value greater than 0. 526 526 Chapter 8 Right Triangles and Trigonometry �� E X A M P L E 4 Using Trigonometric Ratios to Find Lengths Find each length. Round to the nearest hundredth. A AB ̶̶ AB is adjacent to the given angle, ∠A. You are given BC, which is opposite ∠A. Since the adjacent and opposite legs are involved, use a tangent ratio. Do not round until the final step of your answer. Use the values of the trigonometric ratios provided by your calculator. tan A = opp. leg _ adj. leg = BC _ AB Write a trigonometric ratio. tan 41° = 6.1 _ AB AB = 6.1 _ tan 41° AB ≈ 7.02 in. Substitute the given values. Multiply both sides by AB and divide by tan 41°. Simplify the expression. B MP ̶̶̶ MP is opposite the given angle, ∠N. You are given NP, which is the hypotenuse. Since the opposite side and hypotenuse are involved, use a sine ratio. = MP _ NP sin N = opp. leg _ hyp. sin 20° = MP _ 8.7 8.7 (sin 20°)
= MP Write a trigonometric ratio. Substitute the given values. Multiply both sides by 8.7. MP ≈ 2.98 cm Simplify the expression. C YZ YZ is the hypotenuse. You are given XZ, which is adjacent to the given angle, ∠Z. Since the adjacent side and hypotenuse are involved, use a cosine ratio. cos Z = adj. leg _ hyp. cos 38° = 12.6 _ = XZ _ YZ YZ YZ = 12.6 _ cos 38° Write a trigonometric ratio. Substitute the given values. Multiply both sides by YZ and divide by cos 38°. YZ ≈ 15.99 cm Simplify the expression. Find each length. Round to the nearest hundredth. 4a. DF 4b. ST 4c. BC 4d. JL 8-2 Trigonometric Ratios 527 527 �� E X A M P L E 5 Problem Solving Application A contractor is building a wheelchair ramp for a doorway that is 1.2 ft above the ground. To meet ADA guidelines, the ramp will make an angle of 4.8° with the ground. To the nearest hundredth of a foot, what is the horizontal distance covered by the ramp? Understand the Problem Make a sketch. The answer is BC. Make a Plan ̶̶ BC is the leg adjacent to ∠C. You are given AB, which is the leg opposite ∠C. Since the opposite and adjacent legs are involved, write an equation using the tangent ratio. Solve tan C = AB _ BC tan 4.8° = 1.2 _ BC BC = 1.2 _ tan 4.8° Write a trigonometric ratio. Substitute the given values. Multiply both sides by BC and divide by tan 4.8°. BC ≈ 14.2904 ft Simplify the expression. Look Back The problem asks for BC rounded to the nearest hundredth, so round the length to 14.29. The ramp covers a horizontal distance of 14.29 ft. 5. Find AC, the length of the ramp in Example 5, to the nearest hundredth of a foot. THINK AND DISCUSS 1. Tell how you could use a sine ratio to find AB. 2. Tell how you could use a cos
ine ratio to find AB. 3. GET ORGANIZED Copy and complete the graphic organizer. In each cell, write the meaning of each abbreviation and draw a diagram for each. 528 528 Chapter 8 Right Triangles and Trigonometry ge07sec08l02002aABCAB12��34�� 8-2 Exercises Exercises KEYWORD: MG7 8-2 KEYWORD: MG7 Parent GUIDED PRACTICE Vocabulary Apply the vocabulary from this lesson to answer each question. 1. In △JKL, ∠K is a right angle. Write the sine of ∠J as a ratio of side lengths. 2. In △MNP, ∠M is a right angle. Write the tangent of ∠N as a ratio of side lengths. 525 Write each trigonometric ratio as a fraction and as a decimal rounded to the nearest hundredth. 3. sin C 6. cos C 4. tan A 7. tan C 5. cos A 8. sin Use a special right triangle to write each trigonometric ratio as a fraction. p. 526 9. cos 60° 10. tan 30° 11. sin 45. 526 Use your calculator to find each trigonometric ratio. Round to the nearest hundredth. 12. tan 67° 15. cos 88° 13. sin 23° 16. cos 12° 14. sin 49° 17. tan 9 Find each length. Round to the nearest hundredth. p. 527 18. BC 19. QR 20. KL. 528 21. Architecture A pediment has a pitch of 15°, as shown. If the width of the pediment, WZ, is 56 ft, what is XY to the nearest inch? �� Independent Practice Write each trigonometric ratio as a fraction and as a PRACTICE AND PROBLEM SOLVING For See Exercises Example 22–27 28–30 31–36 37–42 43 1 2 3 4 5 TEKS TEKS TAKS TAKS Skills Practice p. S18 Application Practice p. S35 decimal rounded to the nearest hundredth. 22. cos D 25. cos F 23. tan D 26. sin F 24. tan F 27. sin D Use
a special right triangle to write each trigonometric ratio as a fraction. 28. tan 60° 29. sin 30° 30. cos 45° Use your calculator to find each trigonometric ratio. Round to the nearest hundredth. 31. tan 51° 34. tan 14° 32. sin 80° 35. sin 55° 33. cos 77° 36. cos 48° 8-2 Trigonometric Ratios 529 529 �� Find each length. Round to the nearest hundredth. 37. PQ 38. AC 39. GH 40. 40. XZ 41. KL 42. EF 43. Sports A jump ramp for waterskiing makes an angle of 15° with the surface of the water. The ramp rises 1.58 m above the surface. What is the length of the ramp to the nearest hundredth of a meter? �� Use special right triangles to complete each statement. �� 44. An angle that measures 45. For a 45° angle, the 46. The sine of a? has a tangent of 1. ̶̶̶̶? and ̶̶̶̶? angle is 0.5. ̶̶̶̶? ratios are equal. ̶̶̶̶ Sports The Aquaplex Ski Lake in Austin hosted the 2004 and 2005 Barefoot National Championships. Barefoot skiing began in 1947. The first U.S. national competition was held in Waco in September 1978. 47. The cosine of a 30° angle is equal to the sine of a? angle. ̶̶̶̶ 48. Safety According to the Occupational Safety and Health Administration (OSHA), a ladder that is placed against a wall should make a 75.5° angle with the ground for optimal safety. To the nearest tenth of a foot, what is the maximum height that a 10-ft ladder can safely reach? Find the indicated length in each rectangle. Round to the nearest tenth. 49. BC 50. SU 51. Critical Thinking For what angle measures is the tangent ratio less than 1? greater than 1? Explain. 52. This problem will prepare you for the Multi-Step TAKS Prep on page 542. ̶̶ AB at A utility worker is installing a 25-foot pole ̶̶ AD, will help ̶̶ AC and the foot of a hill. Two guy wires, keep the pole vertical. a. To the nearest inch, how long should b. ̶̶
AD is perpendicular to the hill, which makes an angle of 28° with a horizontal line. To the nearest inch, how long should this guy wire be? ̶̶ AC be? 530 530 Chapter 8 Right Triangles and Trigonometry �� 53. Find the sine of the smaller acute angle in a triangle with side lengths of 3, 4, and 5 inches. History 54. Find the tangent of the greater acute angle in a triangle with side lengths of 7, 24, and 25 centimeters. The Pyramid of Cheops consists of more than 2,000,000 blocks of stone with an average weight of 2.5 tons each. 55. History The Great Pyramid of Cheops in Giza, Egypt, was completed around 2566 B.C.E. Its original height was 482 ft. Each face of the pyramid forms a 52° angle with the ground. To the nearest foot, how long is the base of the pyramid? 56. Measurement Follow these steps to calculate trigonometric ratios. a. Use a centimeter ruler to find AB, BC, and AC. b. Use your measurements from part a to find the sine, cosine, and tangent of ∠A. c. Use a protractor to find m∠A. d. Use a calculator to find the sine, cosine, and tangent of ∠A. e. How do the values in part d compare to the ones you found in part b? 57. Algebra Recall from Algebra I that an identity is an equation that is true for all values of the variables. a. Show that the identity tan A = sin A _ is true when m∠A = 30°. cos A b. Write tan A, sin A, and cos A in terms of a, b, and c. c. Use your results from part b to prove the identity tan A = sin A _. cos A Verify that (sin A) 2 + (cos A) 2 = 1 for each angle measure. 58. m∠A = 45° 61. Multi-Step The equation (sin A) 2 + (cos A) 2 = 1 is known as a 59. m∠ A = 30° 60. m∠A = 60° Pythagorean Identity. a. Write sin A and cos A in terms of a, b, and
c. b. Use your results from part a to prove the identity (sin A) 2 + (cos A) 2 = 1. c. Write About It Why do you think this identity is called a Pythagorean identity? Find the perimeter and area of each triangle. Round to the nearest hundredth. 62. 64. 63. 65. 66. Critical Thinking Draw △ABC with ∠C a right angle. Write sin A and cos B in terms of the side lengths of the triangle. What do you notice? How are ∠A and ∠B related? Make a conjecture based on your observations. 67. Write About It Explain how the tangent of an acute angle changes as the angle measure increases. 8-2 Trigonometric Ratios 531 531 �� 68. Which expression can be used to find AB? 7.1 (sin 25°) 7.1 (cos 25°) 7.1 (sin 65°) 7.1 (tan 65°) 69. A steel cable supports an electrical tower as shown. The cable makes a 65° angle with the ground. The base of the cable is 17 ft from the tower. What is the height of the tower to the nearest foot? 8 feet 15 feet 36 feet 40 feet 70. Which of the following has the same value as sin M? sin N tan M cos N cos M CHALLENGE AND EXTEND Algebra Find the value of x. Then find AB, BC, and AC. Round each to the nearest unit. 71. 72. 73. Multi-Step Prove the identity (tan A) 2 + 1 = 1 _. (cos A) 2 74. A regular pentagon with 1 in. sides is inscribed in a circle. Find the radius of the circle rounded to the nearest hundredth. Each of the three trigonometric ratios has a reciprocal ratio, as defined below. These ratios are cosecant (csc), secant (sec), and cotangent (cot). csc A = 1 _ sin A sec A = 1 _ cot A = 1 _ cos A tan A Find each trigonometric ratio to the nearest hundredth. 75. csc Y 76. sec Z 77. cot Y SPIRAL REVIEW Find three ordered pairs that satisfy each function. (Previous course) 78. f (x) = 3x - 6 79. f (x) = -0.
5x + 10 80. f (x) = x 2 - 4x + 2 Identify the property that justifies each statement. (Lesson 2-5) 81. 82. ̶̶ AB ≅ ̶̶ AB ≅ ̶̶ CD, and ̶̶ AB ̶̶ CD ≅ ̶̶ DE. So ̶̶ AB ≅ ̶̶ DE. 83. If ∠JKM ≅ ∠MLK, then ∠MLK ≅ ∠JKM. Find the geometric mean of each pair of numbers. (Lesson 8-1) 84. 3 and 27 85. 6 and 24 86. 8 and 32 532 532 Chapter 8 Right Triangles and Trigonometry �� Inverse Functions Algebra In Algebra, you learned that a function is a relation in which each element of the domain is paired with exactly one element of the range. If you switch the domain and range of a one-to-one function, you create an inverse function. See Skills Bank page S62 The function y = sin -1 x is the inverse of the function y = sin x. �� If you know the value of a trigonometric ratio, you can use the inverse trigonometric function to find the angle measure. You can do this either with a calculator or by looking at the graph of the function. �� Example Use the graphs above to find the value of x for 1 = sin x. Then write this expression using an inverse trigonometric function. 1 = sin x x = 90° Look at the graph of y = sin x. Find where the graph intersects the line y = 1 and read the corresponding x-coordinate. 90° = sin -1 (1) Switch the x- and y-values. Try This TAKS Grades 9–11 Obj. 2 Use the graphs above to find the value of x for each of the following. Then write each expression using an inverse trigonometric function. 1. 0 = sin x 4. 0 = cos x 2. 1 _ 2 = cos x 5. 0 = tan x 3. 1 = tan x 6. 1 _ 2 = sin x On Track for TAKS 533 533 ��
�� 8-3 Solving Right Triangles TEKS G.11.C Similarity and the geometry of shape: develop, apply, and justify triangle similarity relationships, such as... trigonometric ratios.... Objective Use trigonometric ratios to find angle measures in right triangles and to solve real-world problems. Also G.5.D, G.7.A, G.7.C, G.8.C, G.11.B Why learn this? You can convert the percent grade of a road to an angle measure by solving a right triangle. San Francisco, California, is famous for its steep streets. The steepness of a road is often expressed as a percent grade. Filbert Street, the steepest street in San Francisco, has a 31.5% grade. This means the road rises 31.5 ft over a horizontal distance of 100 ft, which is equivalent to a 17.5° angle. You can use trigonometric ratios to change a percent grade to an angle measure. E X A M P L E 1 Identifying Angles from Trigonometric Ratios Use the trigonometric ratio cos A = 0.6 to determine which angle of the triangle is ∠A. cos A = adj. leg_ hyp. cos ∠1 = 3.6_ 6 cos ∠2 = 4.8_ 6 = 0.6 = 0.8 Cosine is the ratio of the adjacent leg to the hypotenuse. The leg adjacent to ∠1 is 3.6. The hypotenuse is 6. The leg adjacent to ∠2 is 4.8. The hypotenuse is 6. Since cos A = cos ∠1, ∠1 is ∠A. Use the given trigonometric ratio to determine which angle of the triangle is ∠A. 1a. sin A = 8 _ 17 1b. tan A = 1.875 In Lesson 8-2, you learned that sin 30° = 0.5. Conversely, if you know that the sine of an acute angle is 0.5, you can conclude that the angle measures 30°. This is written as sin -1 (0.5) = 30°. If you know the sine, cosine, or tangent of an acute angle measure, you can use the inverse trigonometric functions to find the measure of the angle. Inverse Trigonometric Functions If sin A = x, then sin -1
x = m∠A. If cos A = x, then cos -1 x = m∠A. If tan A = x, then tan -1 x = m∠A. The expression sin -1 x is read “the inverse sine of x.” It does not mean 1 ____. You can think of sin -1 x as “the angle whose sine is x.” sin x 534 534 Chapter 8 Right Triangles and Trigonometry �� E X A M P L E 2 Calculating Angle Measures from Trigonometric Ratios Use your calculator to find each angle measure to the nearest degree. A cos -1 (0.5) B sin -1 (0.45) C tan -1 (3.2) When using your calculator to find the value of an inverse trigonometric expression, you may need to press the [arc], [inv], or [2nd] key. cos -1 (0.5) = 60° sin -1 (0.45) ≈ 27° tan -1 (3.2) ≈ 73° Use your calculator to find each angle measure to the nearest degree. 2a. tan -1 (0.75) 2b. cos -1 (0.05) 2c. sin -1 (0.67) Using given measures to find the unknown angle measures or side lengths of a triangle is known as solving a triangle. To solve a right triangle, you need to know two side lengths or one side length and an acute angle measure. E X A M P L E 3 Solving Right Triangles Find the unknown measures. Round lengths to the nearest hundredth and angle measures to the nearest degree. Method 1: Method 2: By the Pythagorean Theorem, AC 2 = AB 2 + BC 2. = (7.5) 2 + 5 2 = 81.25 So AC = √  81.25 ≈ 9.01. m∠A = tan -1 ( 5 _ ) ≈ 34° 7.5 Since the acute angles of a right triangle are complementary, m∠C ≈ 90° - 34° ≈ 56°. m∠A = tan -1 ( 5 _ ) ≈ 34° 7.5 Since the acute angles of a right triangle are complementary, m∠C ≈ 90° - 34°
≈ 56°. sin A = 5 _ AC, so AC = 5 _. sin A 5 __ ≈ 9.01 ⎤ ⎡ tan -1 ( 5 _ ) ⎥ ⎢ sin ⎦ ⎣ AC ≈ 7.5 3. Find the unknown measures. Round lengths to the nearest hundredth and angle measures to the nearest degree. Solving Right Triangles Rounding can really make a difference! To find AC, I used the Pythagorean Theorem and got 15.62. Then I did it a different way. I used m∠A = tan -1 ( 10 __ find m∠A = 39.8056°, which I rounded to 40°. sin 40° = 10 ___ AC, so AC = 10 _____ ≈ 15.56. sin 40° 12 ) to Kendell Waters Marshall High School The difference in the two answers reminded me not to round values until the last step. 8- 3 Solving Right Triangles 535 535 �� E X A M P L E 4 Solving a Right Triangle in the Coordinate Plane The coordinates of the vertices of △JKL are J (-1, 2), K (-1, -3), and L (3, -3). Find the side lengths to the nearest hundredth and the angle measures to the nearest degree. Step 1 Find the side lengths. Plot points J, K, and L. JK = 5 KL = 4 By the Distance Formula, JL = √  ⎤ ⎦ ⎡ ⎣ 2 + (-3 - 2) 2. 3 - (-1) √  4 2 + (-5) 2 = 16 + 25 = √  41 ≈ 6.40 = √  Step 2 Find the angle measures. m∠K = 90° m∠J = tan -1 ( 4 _ ) ≈ 39° 5 ̶̶ KL are ⊥. ̶̶ JK and ̶̶ KL is opp. ∠J, and ̶̶ JK is adj. to ∠J. m∠L ≈ 90°
- 39° ≈ 51° The acute  of a rt. △ are comp. 4. The coordinates of the vertices of △RST are R (-3, 5), S (4, 5), and T (4, -2). Find the side lengths to the nearest hundredth and the angle measures to the nearest degree. E X A M P L E 5 Travel Application San Francisco’s Lombard Street is known as one of “the crookedest streets in the world.” The road’s eight switchbacks were built in the 1920s to make the steep hill passable by cars. If the hill has a percent grade of 84%, what angle does the hill make with a horizontal line? Round to the nearest degree. 84% = 84 _ 100 Change the percent grade to a fraction. An 84% grade means the hill rises 84 ft for every 100 ft of horizontal distance. Draw a right triangle to represent the hill. ∠A is the angle the hill makes with a horizontal line. m∠A = tan -1 ( 84 _ ) ≈ 40° 100 5. Baldwin St. in Dunedin, New Zealand, is the steepest street in the world. It has a grade of 38%. To the nearest degree, what angle does Baldwin St. make with a horizontal line? 536 536 Chapter 8 Right Triangles and Trigonometry �� THINK AND DISCUSS 1. Describe the steps you would use to solve △RST. 2. Given that cos Z = 0.35, write an equivalent statement using an inverse trigonometric function. 3. GET ORGANIZED Copy and complete the graphic organizer. In each box, write a trigonometric ratio for ∠A. Then write an equivalent statement using an inverse trigonometric function. 8-3 Exercises Exercises GUIDED PRACTICE KEYWORD: MG7 8-3 KEYWORD: MG7 Parent. 534 Use the given trigonometric ratio to determine which angle of the triangle is ∠A. 1. sin A = 4 _ 5 2. tan A = 1 1 _ 3 3. cos A = 0.6 4. cos A = 0.8 5. tan A = 0.75 6. sin A = 0. Use your calculator to find each angle measure to the nearest degree. p. 535 7. tan -1 (2.
1) 10. sin -1 (0.5) 8. cos -1 ( 1 _ ) 11. sin -1 (0.61) 3 9. cos -1 ( 5 _ ) 12. tan -1 (0.09. 535 Multi-Step Find the unknown measures. Round lengths to the nearest hundredth and angle measures to the nearest degree. 13. 14. 15. 536 Multi-Step For each triangle, find the side lengths to the nearest hundredth and the angle measures to the nearest degree. 16. D (4, 1), E (4, -2), F (-2, -2) 17. R (3, 3), S (-2, 3), T (-2, -3) 18. X (4, -6), Y (-3, 1), Z (-3, -6) 19. A (-1, 1), B (1, 1), C (1, 5) 8- 3 Solving Right Triangles 537 537 �� 20. Cycling A hill in the Tour de France p. 536 bike race has a grade of 8%. To the nearest degree, what is the angle that this hill makes with a horizontal line? Independent Practice Use the given trigonometric ratio to determine which angle PRACTICE AND PROBLEM SOLVING For See Exercises Example 21–26 27–32 33–35 36–37 38 1 2 3 4 5 TEKS TEKS TAKS TAKS Skills Practice p. S18 Application Practice p. S35 of the triangle is ∠A. 21. tan A = 5 _ 12 24. sin A = 5 _ 13 22. tan A = 2.4 25. cos A = 12 _ 13 23. sin A = 12 _ 13 26. cos A = 5 _ 13 Use your calculator to find each angle measure to the nearest degree. 27. sin -1 (0.31) 27. 28. tan -1 (1) 30. cos -1 (0.72) 31. tan -1 (1.55) 29. cos -1 (0.8) 32. sin -1 ( 9 _ ) 17 Multi-Step Find the unknown measures. Round lengths to the nearest hundredth and angle measures to the nearest degree. 33. 34
. 35. Multi-Step For each triangle, find the side lengths to the nearest hundredth and the angle measures to the nearest degree. 36. A (2, 0), B (2, -5), C (1, -5) 37. M (3, 2), N (3, -2), P (-1, -2) 38. Building For maximum accessibility, a wheelchair ramp should have a slope between 1 __ 16 and 1 __ 20. What is the range of angle measures that a ramp should make with a horizontal line? Round to the nearest degree. Complete each statement. If necessary, round angle measures to the nearest degree. Round other values to the nearest hundredth.? ≈ 3.5 ̶̶̶̶ 39. tan 42. cos -1 ( 45. Critical Thinking Use trigonometric ratios to explain why the diagonal of? 42° ≈ 0.74 ̶̶̶̶? 60° = 1 _ ̶̶̶̶ 2 40. sin 43. sin -1 (? ≈ 2 _ ̶̶̶̶ 3? ) ≈ 69° ̶̶̶̶? ) ≈ 12° ̶̶̶̶ 41. 44. a square forms a 45° angle with each of the sides. 46. Estimation You can use trigonometry to find angle measures when a protractor is not available. a. Estimate the measure of ∠P. b. Use a centimeter ruler to find RQ and PQ. c. Use your measurements from part b and an inverse trigonometric function to find m∠P to the nearest degree. d. How does your result in part c compare to your estimate in part a? 538 538 Chapter 8 Right Triangles and Trigonometry �� 46. This problem will prepare you for the Multi-Step TAKS Prep on page 542. An electric company wants to install a vertical utility pole at the base of a hill that has an 8% grade. a. To the nearest degree, what angle does the hill make with a horizontal line? b. What is the measure of the angle between the pole and the hill? Round to the nearest degree. c. A utility worker installs a 31-foot guy wire from the top of the pole to the hill. Given that the guy wire is perpendicular to the hill, find the height of the pole to the nearest inch
. The side lengths of a right triangle are given below. Find the measures of the acute angles in the triangle. Round to the nearest degree. 48. 3, 4, 5 49. 5, 12, 13 50. 8, 15, 17 51. What if…? A right triangle has leg lengths of 28 and 45 inches. Suppose the length of the longer leg doubles. What happens to the measure of the acute angle opposite that leg? 52. Fitness As part of off-season training, the Houston Texans football team must sprint up a ramp with a 28% grade. To the nearest degree, what angle does this ramp make with a horizontal line? 53. The coordinates of the vertices of a triangle are A (-1, 0), B (6, 1), and C (0, 3). a. Use the Distance Formula to find AB, BC, and AC. b. Use the Converse of the Pythagorean Theorem to show that △ABC is a right triangle. Identify the right angle. c. Find the measures of the acute angles of △ABC. Round to the nearest degree. Find the indicated measure in each rectangle. Round to the nearest degree. 54. m∠BDC 55. m∠STV Find the indicated measure in each rhombus. Round to the nearest degree. 56. m∠DGF 57. m∠LKN Fitness Running on a treadmill is slightly easier than running outdoors, since you don’t have to overcome wind resistance. Set the treadmill to a 1% grade to match the intensity of an outdoor run. 58. Critical Thinking Without using a calculator, compare the values of tan 60° and tan 70°. Explain your reasoning. The measure of an acute angle formed by a line with slope m and the x-axis can be found by using the expression tan -1 (m). Find the measure of the acute angle that each line makes with the x-axis. Round to the nearest degree. 59. y = 3x + 5 60. y = 2 _ 3 x + 1 61. 5y = 4x + 3 8- 3 Solving Right Triangles 539 539 �� 62. /////ERROR ANALYSIS///// A student was asked to find m∠C. Explain the error in the student’s solution. 63. Write About It
A student claims that you must know the three side lengths of a right triangle before you can use trigonometric ratios to find the measures of the acute angles. Do you agree? Why or why not? ̶̶ DC is an altitude of right △ABC. Use trigonometric ratios to find the missing lengths in the figure. Then use these lengths to verify the three relationships in the Geometric Mean Corollaries from Lesson 8-1. 64. 65. Which expression can be used to find m∠A? tan -1 (0.75) sin -1 ( 3 _ ) 5 cos -1 (0.8) tan -1 ( 4 _ ) 3 66. Which expression is NOT equivalent to cos 60°? 1 _ 2 sin 30° sin 60° _ tan 60° cos -1 ( 1 _ ) 2 67. To the nearest degree, what is the measure of the acute angle formed by Jefferson St. and Madison St.? 27° 31° 59° 63° 68. Gridded Response A highway exit ramp. To the nearest degree, has a slope of 3 __ 20 find the angle that the ramp makes with a horizontal line. CHALLENGE AND EXTEND Find each angle measure. Round to the nearest degree. 69. m∠J 70. m∠A Simply each expression. 71. cos -1 (cos 34°) ⎤ ⎦ ⎡ ⎣ tan -1 (1.5) 72. tan 73. sin ( sin -1 x) 74. A ramp has a 6% grade. The ramp is 40 ft long. Find the vertical distance that the ramp rises. Round your answer to the nearest hundredth. 540 540 Chapter 8 Right Triangles and Trigonometry ��Main St.Jefferson St.Madison St.2.7 mi1.4 mige07se_c08l03007aAB�� 75. Critical Thinking Explain why the expression sin -1 (1.5) does not make sense. 76. If you are given the lengths of two sides of △ABC and the measure of the included angle, you can use the formula 1 __ 2 bc sin A to find the area of the triangle. Derive this formula. (Hint: Draw an altitude from B to ratios to find the length
of this altitude.) ̶̶ AC. Use trigonometric SPIRAL REVIEW The graph shows the amount of rainfall in a city for the first five months of the year. Determine whether each statement is true or false. (Previous course) 77. It rained more in April than it did in January, February, and March combined. 78. The average monthly rainfall for this five- month period was approximately 3.5 inches. 79. The rainfall amount increased at a constant rate each month over the five-month period. Use the diagram to find each value, given that △ABC ≅ △DEF. (Lesson 4-3) 80. x 81. y 82. DF Use your calculator to find each trigonometric ratio. Round to the nearest hundredth. (Lesson 8-2) 83. sin 63° 84. cos 27° 85. tan 64° Using Technology Use a spreadsheet to complete the following. 1. In cells A2 and B2, enter values for the leg lengths of a right triangle. 2. In cell C2, write a formula to calculate c, the length of the hypotenuse. 3. Write a formula to calculate the measure of ∠A in cell D2. Be sure to use the Degrees function so that the answer is given in degrees. Format the value to include no decimal places. 4. Write a formula to calculate the measure of ∠B in cell E2. Again, be sure to use the Degrees function and format the value to include no decimal places. 5. Use your spreadsheet to check your answers for Exercises 48–50. 8- 3 Solving Right Triangles 541 541 �� SECTION 8A Trigonometric Ratios It’s Electrifying! Utility workers install and repair the utility poles and wires that carry electricity from generating stations to consumers. As shown in the figure, a crew of workers ̶̶ AC plans to install a vertical utility pole ̶̶ AB that is and a supporting guy wire perpendicular to the ground. 1. The utility pole is 30 ft tall. The crew finds that DC = 6 ft. What is the distance DB from the pole to the anchor point of the guy wire? 2. How long is the guy wire? Round to the nearest inch. 3
. In the figure, ∠ABD is called the line angle. In order to choose the correct weight of the cable for the guy wire, the crew needs to know the measure of the line angle. Find m∠ABD to the nearest degree. 4. To the nearest degree, what is the measure of the angle formed by the pole and the guy wire? 5. What is the percent grade of the hill on which the crew is working? 542 542 Chapter 8 Right Triangles and Trigonometry �� SECTION 8A Quiz for Lessons 8-1 Through 8-3 8-1 Similarity in Right Triangles Find the geometric mean of each pair of numbers. If necessary, give the answer in simplest radical form. 1. 5 and 12 2. 2.75 and 44 and 15 _ 3. 5 _ 8 2 Find x, y, and z. 4. 5. 6. 7. A land developer needs to know the distance across a pond on a piece of property. What is AB to the nearest tenth of a meter? �� 8-2 Trigonometric Ratios Use a special right triangle to write each trigonometric ratio as a fraction. 8. tan 45° 9. sin 30° 10. cos 30° Use your calculator to find each trigonometric ratio. Round to the nearest hundredth. 11. sin 16° 12. cos 79° 13. tan 27° Find each length. Round to the nearest hundredth. 14. QR 15. AB 16. LM 8-3 Solving Right Triangles Find the unknown measures. Round lengths to the nearest hundredth and angle measures to the nearest degree. 17. 18. 19. 20. The wheelchair ramp at the entrance of the Mission Bay Library has a slope of 1 __ 18. What angle does the ramp make with the sidewalk? Round to the nearest degree. Ready to Go On? 543 543 �� 8-4 Angles of Elevation and Depression TEKS G.11.C Similarity and the geometry of shape:... apply... triangle similarity relationships, such as... trigonometric ratios.... Also G.5.D Objective Solve problems involving angles of elevation and angles of depression. Who uses this? Pilots and air traffic controllers use angles of depression to calculate distances. Vocabulary angle of elevation angle of depression An angle of elevation is the angle formed by
a horizontal line and a line of sight to a point above the line. In the diagram, ∠1 is the angle of elevation from the tower T to the plane P. An angle of depression is the angle formed by a horizontal line and a line of sight to a point below the line. ∠2 is the angle of depression from the plane to the tower. Since horizontal lines are parallel, ∠1 ≅ ∠2 by the Alternate Interior Angles Theorem. Therefore the angle of elevation from one point is congruent to the angle of depression from the other point. E X A M P L E 1 Classifying Angles of Elevation and Depression Classify each angle as an angle of elevation or angle of depression. A ∠3 ∠3 is formed by a horizontal line and a line of sight to a point below the line. It is an angle of depression. B ∠4 ∠4 is formed by a horizontal line and a line of sight to a point above the line. It is an angle of elevation. Use the diagram above to classify each angle as an angle of elevation or angle of depression. 1a. ∠5 1b. ∠6 544 544 Chapter 8 Right Triangles and Trigonometry PT12ge07se_c08l04002aABAngle of depressionAngle of elevation3465ge07se_c08l04006aAB E X A M P L E 2 Finding Distance by Using Angle of Elevation An air traffic controller at an airport sights a plane at an angle of elevation of 41°. The pilot reports that the plane’s altitude is 4000 ft. What is the horizontal distance between the plane and the airport? Round to the nearest foot. Draw a sketch to represent the given information. Let A represent the airport and let P represent the plane. Let x be the horizontal distance between the plane and the airport. tan 41° = 4000 _ x x = 4000 _ tan 41° You are given the side opposite ∠A, and x is the side adjacent to ∠A. So write a tangent ratio. Multiply both sides by x and divide both sides by tan 41°. x ≈ 4601 ft Simplify the expression. 2. What if…? Suppose the plane is at an altitude of 3500 ft and the angle of elevation from the airport to the plane is 29°. What is the horizontal distance between the plane and the airport? Round to the nearest
foot. E X A M P L E 3 Finding Distance by Using Angle of Depression A forest ranger in a 90-foot observation tower sees a fire. The angle of depression to the fire is 7°. What is the horizontal distance between the tower and the fire? Round to the nearest foot. Draw a sketch to represent the given information. Let T represent the top of the tower and let F represent the fire. Let x be the horizontal distance between the tower and the fire. The angle of depression may not be one of the angles in the triangle you are solving. It may be the complement of one of the angles in the triangle. By the Alternate Interior Angles Theorem, m∠F = 7°. tan 7° = 90 _ x Write a tangent ratio. x = 90 _ tan 7° Multiply both sides by x and divide both sides by tan 7°. x ≈ 733 ft Simplify the expression. 3. What if…? Suppose the ranger sees another fire and the angle of depression to the fire is 3°. What is the horizontal distance to this fire? Round to the nearest foot. 8- 4 Angles of Elevation and Depression 545 545 �� E X A M P L E 4 Aviation Application A pilot flying at an altitude of 2.7 km sights two control towers directly in front of her. The angle of depression to the base of one tower is 37°. The angle of depression to the base of the other tower is 58°. What is the distance between the two towers? Round to the nearest tenth of a kilometer. Step 1 Draw a sketch. Let P represent the plane and let A and B represent the two towers. Let x be the distance between the towers. Always make a sketch to help you correctly place the given angle measure. Step 2 Find y. By the Alternate Interior Angles Theorem, m∠CAP = 58°. In △APC, tan 58° = 2.7 _ y. So y = 2.7 _ ≈ 1.6871 km. tan 58° Step 3 Find z. By the Alternate Interior Angles Theorem, m∠CBP = 37°. In △BPC, tan 37° = 2.7 _ z. So z = 2.7 _ ≈ 3.5830 km. tan 37° Step 4 Find x. x = z - y x ≈ 3.5830 - 1.6871 ≈
1.9 km So the two towers are about 1.9 km apart. 4. A pilot flying at an altitude of 12,000 ft sights two airports directly in front of him. The angle of depression to one airport is 78°, and the angle of depression to the second airport is 19°. What is the distance between the two airports? Round to the nearest foot. THINK AND DISCUSS 1. Explain what happens to the angle of elevation from your eye to the top of a skyscraper as you walk toward the skyscraper. 2. GET ORGANIZED Copy and complete the graphic organizer below. In each box, write a definition or make a sketch. 546 546 Chapter 8 Right Triangles and Trigonometry �� 8-4 Exercises Exercises KEYWORD: MG7 8-4 KEYWORD: MG7 Parent GUIDED PRACTICE Vocabulary Apply the vocabulary from this lesson to answer each question. 1. An angle of? is measured from a horizontal line to a point above that line. ̶̶̶̶ (elevation or depression) 2. An angle of? is measured from a horizontal line to a point below that line. ̶̶̶̶ (elevation or depression. 544 Classify each angle as an angle of elevation or angle of depression. 3. ∠1 4. ∠2 5. ∠3 6. ∠. Measurement When the angle of elevation to p. 545 the sun is 37°, a flagpole casts a shadow that is 24.2 ft long. What is the height of the flagpole to the nearest foot. 545 8. Aviation The pilot of a traffic helicopter sights an accident at an angle of depression of 18°. The helicopter’s altitude is 1560 ft. What is the horizontal distance from the helicopter to the accident? Round to the nearest foot. 546 9. Surveying From the top of a canyon, the angle of depression to the far side of the river is 58°, and the angle of depression to the near side of the river is 74°. The depth of the canyon is 191 m. What is the width of the river at the bottom of the canyon? Round to the nearest tenth of a meter. �� Independent Practice For See Exercises Example 10–13 14 15 16 1 2 3 4
TEKS TEKS TAKS TAKS Skills Practice p. S19 Application Practice p. S35 PRACTICE AND PROBLEM SOLVING Classify each angle as an angle of elevation or angle of depression. 10. ∠1 11. ∠2 12. ∠3 13. ∠4 14. Geology To measure the height of a rock formation, a surveyor places her transit 100 m from its base and focuses the transit on the top of the formation. The angle of elevation is 67°. The transit is 1.5 m above the ground. What is the height of the rock formation? Round to the nearest meter. 8- 4 Angles of Elevation and Depression 547 547 1234ge07se_c08l04008a AB1423ge07sec08l04007a�� Space Shuttle Johnson Space Center, in Houston, is home to the Mission Control Center, the base of operations for all space shuttle missions. 15. Forestry A forest ranger in a 120 ft observation tower sees a fire. The angle of 15. depression to the fire is 3.5°. What is the horizontal distance between the tower and the fire? Round to the nearest foot. 16. Space Shuttle Marion is observing the launch of a space shuttle from the command center. When she first sees the shuttle, the angle of elevation to it is 16°. Later, the angle of elevation is 74°. If the command center is 1 mi from the launch pad, how far did the shuttle travel while Marion was watching? Round to the nearest tenth of a mile. Tell whether each statement is true or false. If false, explain why. 17. The angle of elevation from your eye to the top of a tree increases as you walk toward the tree. 18. If you stand at street level, the angle of elevation to a building’s tenth-story window is greater than the angle of elevation to one of its ninth-story windows. 19. As you watch a plane fly above you, the angle of elevation to the plane gets closer to 0° as the plane approaches the point directly overhead. 20. An angle of depression can never be more than 90°. � � � � Use the diagram for Exercises 21 and 22. 21. Which angles are not angles of elevation or angles of depression? 22. The angle of depression from the helicopter to the car is 30°. Find m∠1, m∠2, m∠
3, and m∠4. 23. Critical Thinking Describe a situation in which the angle of depression to an object is decreasing. 24. An observer in a hot-air balloon sights a building that is 50 m from the balloon’s launch point. The balloon has risen 165 m. What is the angle of depression from the balloon to the building? Round to the nearest degree. 25. Multi-Step A surveyor finds that the angle of elevation to the top of a 1000 ft tower is 67°. a. To the nearest foot, how far is the surveyor from the base of the tower? b. How far back would the surveyor have to move so that the angle of elevation to the top of the tower is 55°? Round to the nearest foot. 26. Write About It Two students are using shadows to calculate the height of a pole. One says that it will be easier if they wait until the angle of elevation to the sun is exactly 45°. Explain why the student made this suggestion. 27. This problem will prepare you for the Multi-Step TAKS Prep on page 568. The pilot of a rescue helicopter is flying over the ocean at an altitude of 1250 ft. The pilot sees a life raft at an angle of depression of 31°. a. What is the horizontal distance from the helicopter to the life raft, rounded to the nearest foot? b. The helicopter travels at 150 ft/s. To the nearest second, how long will it take until the helicopter is directly over the raft? 548 548 Chapter 8 Right Triangles and Trigonometry 16º74ºge07sec08l04003_A1 mi �� 28. Mai is flying a plane at an altitude of 1600 ft. She sights a stadium at an angle of depression of 35°. What is Mai’s approximate horizontal distance from the stadium? 676 feet 1120 feet 1450 feet 2285 feet 29. Jeff finds that an office building casts a shadow that is 93 ft long when the angle of elevation to the sun is 60°. What is the height of the building? 54 feet 81 feet 107 feet 161 feet 30. Short Response Jim is rafting down a river that runs through a canyon. He sees a trail marker ahead at the top of the canyon and estimates the angle of elevation from the raft to the marker as 45°. Draw a sketch to represent the situation. Explain what happens to the angle of elevation as Jim moves closer to the marker. CH
ALLENGE AND EXTEND 31. Susan and Jorge stand 38 m apart. From Susan’s position, the angle of elevation to the top of Big Ben is 65°. From Jorge’s position, the angle of elevation to the top of Big Ben is 49.5°. To the nearest meter, how tall is Big Ben? �� 32. A plane is flying at a constant altitude of 14,000 ft and a constant speed of 500 mi/h. The angle of depression from the plane to a lake is 6°. To the nearest minute, how much time will pass before the plane is directly over the lake? 33. A skyscraper stands between two school buildings. The two schools are 10 mi apart. From school A, the angle of elevation to the top of the skyscraper is 5°. From school B, the angle of elevation is 2°. What is the height of the skyscraper to the nearest foot? 34. Katie and Kim are attending a theater performance. Katie’s seat is at floor level. She looks down at an angle of 18° to see the orchestra pit. Kim’s seat is in the balcony directly above Katie. Kim looks down at an angle of 42° to see the pit. The horizontal distance from Katie’s seat to the pit is 46 ft. What is the vertical distance between Katie’s seat and Kim’s seat? Round to the nearest inch. SPIRAL REVIEW 35. Emma and her mother jog along a mile-long circular path in opposite directions. They begin at the same place and time. Emma jogs at a pace of 4 mi/h, and her mother runs at 6 mi/h. In how many minutes will they meet? (Previous course) 36. Greg bought a shirt that was discounted 30%. He used a coupon for an additional 15% discount. What was the original price of the shirt if Greg paid $17.85? (Previous course) Tell which special parallelograms have each given property. (Lesson 6-5) 37. The diagonals are perpendicular. 38. The diagonals are congruent. 39. The diagonals bisect each other. 40. Opposite angles are congruent. Find each length. (Lesson 8-1) 41. x 42. y 43. z 8- 4 Angles of Elevation and Depression 549 549 �� 8-4 Indirect Measurement
Using Trigonometry A clinometer is a surveying tool that is used to measure angles of elevation and angles of depression. In this lab, you will make a simple clinometer and use it to find indirect measurements. Choose a tall object, such as a flagpole or tree, whose height you will measure. TEKS G.11.C Similarity and the geometry of shape:... apply... triangle similarity relationships, such as... trigonometric ratios.... Also G.5.D, G.11.B Use with Lesson 8-4 Activity 1 Follow these instructions to make a clinometer. a. Tie a washer or paper clip to the end of a 6-inch string. b. Tape the string’s other end to the midpoint of the straight edge of a protractor. c. Tape a straw along the straight edge of the protractor. 2 Stand back from the object you want to measure. Use a tape measure to measure and record the distance from your feet to the base of the object. Also measure the height of your eyes above the ground. 3 Hold the clinometer steady and look through the straw to sight the top of the object you are measuring. When the string stops moving, pinch it against the protractor and record the acute angle measure. Try This 1. How is the angle reading from the clinometer related to the angle of elevation from your eye to the top of the object you are measuring? 2. Draw and label a diagram showing the object and the measurements you made. Then use trigonometric ratios to find the height of the object. 3. Repeat the activity, measuring the angle of elevation to the object from a different distance. How does your result compare to the previous one? 4. Describe possible measurement errors that can be made in the activity. 5. Explain why this method of indirect measurement is useful in real-world situations. 550 550 Chapter 8 Right Triangles and Trigonometry 8-5 Law of Sines and Law of Cosines TEKS G.11.C Similarity and the geometry of shape: develop, apply, and justify triangle similarity relationships, such as... trigonometric ratios.... Also G.5.B, G.5.D, G.7.A, G.11.A Objective Use the Law of Sines and the Law of Cosines to solve triangles. Who uses this? Engineers can use the Law of Sines and the Law of Cosines to solve construction problems. Since its completion in 1370
, engineers have proposed many solutions for lessening the tilt of the Leaning Tower of Pisa. The tower does not form a right angle with the ground, so the engineers have to work with triangles that are not right triangles. In this lesson, you will learn to solve any triangle. To do so, you will need to calculate trigonometric ratios for angle measures up to 180°. You can use a calculator to find these values. E X A M P L E 1 Finding Trigonometric Ratios for Obtuse Angles You will learn more about trigonometric ratios of angle measures greater than or equal to 90° in the Chapter Extension. Use a calculator to find each trigonometric ratio. Round to the nearest hundredth. A sin 135° B tan 98° C cos 108° sin 135° ≈ 0.71 tan 98° ≈ -7.12 cos 108° ≈ -0.31 Use a calculator to find each trigonometric ratio. Round to the nearest hundredth. 1a. tan 175° 1c. sin 160° 1b. cos 92° You can use the altitude of a triangle to find a relationship between the triangle’s side lengths. In △ABC, let h represent the length of the ̶̶ AB. altitude from C to, and sin B = h _ a. From the diagram, sin A = h _ b By solving for h, you find that h = b sin A and h = a sin B. So b sin A = a sin B, and sin A _ a You can use another altitude to show that these ratios equal sin C _. c = sin B _. b 8-5 Law of Sines and Law of Cosines 551 551 �� Theorem 8-5-1 The Law of Sines For any △ABC with side lengths a, b, and c, sin A _ a = sin C _ c = sin B _. b You can use the Law of Sines to solve a triangle if you are given • two angle measures and any side length (ASA or AAS) or • two side lengths and a non-included angle measure (SSA). E X A M P L E 2 Using the Law of Sines Find each measure. Round lengths to the nearest tenth and angle measures to the nearest degree. A DF sin D _ EF sin 105° _ 18 = sin E _ DF = sin 32° _ DF Law of Sines Substitute the
given values. DF sin 105° = 18 sin 32° Cross Products Property In a proportion with three parts, you can use any of the two parts together. DF = 18 sin 32° _ ≈ 9.9 sin 105° B m∠S sin T _ RS sin 75° _ 7 = sin S _ RT = sin S _ 5 sin S = 5 sin 75° _ m∠S ≈ sin -1 ( 5 sin 75° _ 7 7 ) ≈ 44° Divide both sides by sin 105°. Law of Sines Substitute the given values. Multiply both sides by 5. Use the inverse sine function to find m∠S. Find each measure. Round lengths to the nearest tenth and angle measures to the nearest degree. 2a. NP 2b. m∠L 2c. m∠X 2d. AC The Law of Sines cannot be used to solve every triangle. If you know two side lengths and the included angle measure or if you know all three side lengths, you cannot use the Law of Sines. Instead, you can apply the Law of Cosines. 552 552 Chapter 8 Right Triangles and Trigonometry �� Theorem 8-5-2 The Law of Cosines For any △ABC with side lengths a, b, and c: a 2 = b 2 + c 2 - 2bc cos A b 2 = a 2 + c 2 - 2ac cos B c 2 = a 2 + b 2 - 2ab cos C The angle referenced in the Law of Cosines is across the equal sign from its corresponding side. You will prove one case of the Law of Cosines in Exercise 57. You can use the Law of Cosines to solve a triangle if you are given • two side lengths and the included angle measure (SAS) or • three side lengths (SSS). E X A M P L E 3 Using the Law of Cosines Find each measure. Round lengths to the nearest tenth and angle measures to the nearest degree. A BC BC 2 = AB 2 + AC 2 - 2 (AB) (AC) cos A = 14 2 + 9 2 - 2 (14 ) (9 ) cos 62° Law of Cosines Substitute the given BC 2 ≈ 158.6932 BC ≈ 12.6 values. Simplify. Find the square root of both sides. B m∠R ST
2 = RS 2 + RT 2 - 2 (RS) (RT) cos R Law of 9 2 = 4 2 + 7 2 -2 (4 ) (7 ) cos R 81 = 65 - 56 cos R 16 = -56 cos R cos R = - 16 _ 56 m∠R = cos -1 (- 16 _ 56 ) ≈ 107° Cosines Substitute the given values. Simplify. Subtract 65 from both sides. Solve for cos R. Use the inverse cosine function to find m∠R. Find each measure. Round lengths to the nearest tenth and angle measures to the nearest degree. 3a. DE 3b. m∠K 3c. YZ 3d. m∠R 8-5 Law of Sines and Law of Cosines 553 553 �� E X A M P L E 4 Engineering Application The Leaning Tower of Pisa is 56 m tall. In 1999, the tower made a 100° angle with the ground. To stabilize the tower, an engineer considered attaching a cable from the top of the tower to a point that is 40 m from the base. How long would the cable be, and what angle would it make with the ground? Round the length to the nearest tenth and the angle measure to the nearest degree. Step 1 Find the length of the cable. AC 2 = AB 2 + BC 2 - 2 (AB) (BC) cos B = 40 2 + 56 2 - 2 (40 ) (56 ) cos 100° AC 2 ≈ 5513.9438 AC ≈ 74.3 m Law of Cosines Substitute the given values. Simplify. Find the square root of both sides. Do not round your answer until the final step of the computation. If a problem has multiple steps, store the calculated answers to each part in your calculator. Step 2 Find the measure of the angle the cable would make with the ground. sin A _ = sin B _ BC AC sin A _ ≈ sin 100° _ 56 74.2559 sin A ≈ 56 sin 100° _ m∠A ≈ sin -1 ( 56 sin 100° _ 74.2559 74.2559 ) ≈ 48° Law of Sines Substitute the calculated value for AC. Multiply both sides by 56. Use the inverse sine function to find m∠A. 4. What if…? Another engineer suggested
using a cable attached from the top of the tower to a point 31 m from the base. How long would this cable be, and what angle would it make with the ground? Round the length to the nearest tenth and the angle measure to the nearest degree. THINK AND DISCUSS 1. Tell what additional information, if any, is needed to find BC using the Law of Sines. 2. GET ORGANIZED Copy and complete the graphic organizer. Tell which law you would use to solve each given triangle and then draw an example. 554 554 Chapter 8 Right Triangles and Trigonometry 100º56 m40 mABCge07sec08l05002a�� 8-5 Exercises Exercises KEYWORD: MG7 8-5 KEYWORD: MG7 Parent GUIDED PRACTICE Use a calculator to find each trigonometric ratio. Round to the nearest hundredth. p. 551 1. sin 100° 4. tan 141° 7. sin 147° 2. cos 167° 5. cos 133° 8. tan 164° 3. tan 92° 6. sin 150° 9. cos 156. 552 Find each measure. Round lengths to the nearest tenth and angle measures to the nearest degree. 10. RT 11. m∠B 12. m∠ 13. m∠Q 14. MN 15. AB p. 553. 554 16. Carpentry A carpenter makes a triangular frame by joining three pieces of wood that are 20 cm, 24 cm, and 30 cm long. What are the measures of the angles of the triangle? Round to the nearest degree. Independent Practice Use a calculator to find each trigonometric ratio. Round to the nearest hundredth. PRACTICE AND PROBLEM SOLVING For See Exercises Example 17–25 26–31 32–37 38 1 2 3 4 TEKS TEKS TAKS TAKS Skills Practice p. S19 Application Practice p. S35 17. cos 95° 20. sin 132° 23. tan 139° 18. tan 178° 21. sin 98° 24. cos 145° 19. tan 118° 22. cos 124° 25. sin 128° Find each measure. Round lengths to the nearest tenth and
angle measures to the nearest degree. 26. m∠C 27. PR 28. JL 29. EF 30. m∠J 31. m∠X 8-5 Law of Sines and Law of Cosines 555 555 �� Surveying Many modern surveys are done with GPS (Global Positioning System) technology. GPS uses orbiting satellites as reference points from which other locations are established. Find each measure. Round lengths to the nearest tenth and angle measures to the nearest degree. 32. AB 33. m∠Z 34. m∠R 35. EF 36. LM 37. m∠G 38. Surveying To find the distance across a lake, a surveyor locates points A, B, and C as shown. What is AB to the nearest tenth of a meter, and what is m∠B to the nearest degree? Use the figure for Exercises 39–42. Round lengths to the nearest tenth and angle measures to the nearest degree. 39. m∠A = 74°, m∠B = 22°, and b = 3.2 cm. Find a. 40. m∠C = 100°, a = 9.5 in., and b = 7.1 in. Find c. 41. a = 2.2 m, b = 3.1 m, and c = 4 m. Find m∠B. 42. a = 10.3 cm, c = 8.4 cm, and m∠A = 45°. Find m∠C. 43. Critical Thinking Suppose you are given the three angle measures of a triangle. Can you use the Law of Sines or the Law of Cosines to find the lengths of the sides? Why or why not? 44. What if…? What does the Law of Cosines simplify to when the given angle is a right angle? 45. Orienteering The map of a beginning orienteering course is shown at right. To the nearest degree, at what angle should a team turn in order to go from the first checkpoint to the second checkpoint? Multi-Step Find the perimeter of each triangle. Round to the nearest tenth. 46. 47. 48. 49. The ambiguous case of the Law of Sines occurs when you are given an acute angle measure and when the side opposite this angle is shorter than the
other given side. In this case, there are two possible triangles. Find two possible values for m∠C to the nearest degree. (Hint: The inverse sine function on your calculator gives you only acute angle measures. Consider this angle and its supplement.) 556 556 Chapter 8 Right Triangles and Trigonometry 6 km4 km3 kmSecondcheckpointFirstcheckpointStart?ge07se_c08L05003atopo mapGeometry 2007 SEHolt Rinehart WinstonKaren Minot(415)883-6560�� 50. This problem will prepare you for the Multi-Step TAKS Prep on page 568. Rescue teams at two heliports, A and B, receive word of a fire at F. a. What is m∠AFB? b. To the nearest mile, what are the distances from each heliport to the fire? c. If a helicopter travels 150 mi/h, how much time is saved by sending a helicopter from A rather than B? Identify whether you would use the Law of Sines or Law of Cosines as the first step when solving the given triangle. 51. 52. 53. 54. The coordinates of the vertices of △RST are R (0, 3), S (3, 1), and T (-3, -1). a. Find RS, ST, and RT. b. Which angle of △RST is the largest? Why? c. Find the measure of the largest angle in △RST to the nearest degree. 55. Art Jessika is creating a pattern for a piece of stained glass. Find BC, AB, and m∠ABC. Round lengths to the nearest hundredth and angle measures to the nearest degree. 56. /////ERROR ANALYSIS///// Two students were asked to find x in △DEF. Which solution is incorrect? Explain the error. 57. Complete the proof of the Law of Cosines for the case when △ABC is an acute triangle. Given: △ABC is acute with side lengths a, b, and c. Prove: a 2 = b 2 + c 2 - 2bc cos A Proof: Draw the altitude from C to ̶̶ AB. Let h be the length of this
altitude. ̶̶ AB into segments of lengths x and y. By the Pythagorean Theorem, It divides a 2 = a. to get c. expression for b 2 to get d. Therefore a 2 = b 2 + c 2 - 2bc cos A by f.?, and b. ̶̶̶̶?. Rearrange the terms to get - 2cx. Substitute the ̶̶̶̶? = h 2 + x 2. Substitute y = c - x into the first equation ̶̶̶̶?. From the diagram, cos A = x __. So x = e. ̶̶̶̶ b?. ̶̶̶̶?. ̶̶̶̶ 58. Write About It Can you use the Law of Sines to solve △EFG? Explain why or why not. 8-5 Law of Sines and Law of Cosines 557 557 �� 59. Which of these is closest to the length of ̶̶ AB? 5.5 centimeters 7.5 centimeters 14.4 centimeters 22.2 centimeters 60. Which set of given information makes it possible to find x using the Law of Sines? m∠T = 38°, RS = 8.1, ST = 15.3 RS = 4, m∠S = 40°, ST = 9 m∠R = 92°, m∠S = 34°, ST = 7 m∠R = 105°, m∠S = 44°, m∠T = 31° 61. A surveyor finds that the face of a pyramid makes a 135° angle with the ground. From a point 100 m from the base of the pyramid, the angle of elevation to the top is 25°. ̶̶ XY? How long is the face of the pyramid, 48 meters 81 meters 124 meters 207 meters CHALLENGE AND EXTEND 62. Multi-Step Three circular disks are placed next to each other as shown. The disks
have radii of 2 cm, 3 cm, and 4 cm. The centers of the disks form △ABC. Find m∠ACB to the nearest degree. 63. Line ℓ passes through points (-1, 1) and (1, 3). Line m passes through points (-1, 1) and (3, 2). Find the measure of the acute angle formed by ℓ and m to the nearest degree. 64. Navigation The port of Bonner is 5 mi due south of the port of Alston. A boat leaves the port of Alston at a bearing of N 32° E and travels at a constant speed of 6 mi/h. After 45 minutes, how far is the boat from the port of Bonner? Round to the nearest tenth of a mile. SPIRAL REVIEW Write a rule for the nth term in each sequence. (Previous course) 65. 3, 6, 9, 12, 15, … 66. 3, 5, 7, 9, 11, … 67. 4, 6, 8, 10, 12, … State the theorem or postulate that justifies each statement. (Lesson 3-2) 68. ∠1 ≅ ∠8 69. ∠4 ≅ ∠5 70. m∠4 + m∠6 = 180° 71. ∠2 ≅ ∠7 Use the given trigonometric ratio to determine which angle of the triangle is ∠A. (Lesson 8-3) 73. sin A = 15 _ 72. cos A = 15 _ 17 17 74. tan A = 1.875 558 558 Chapter 8 Right Triangles and Trigonometry ��YX135°25°100 mge07se_c08l05005aA2 cm4 cm3 cmBCge07se_c08l05006aaAB�� 8-6 Vectors TEKS G.7.A Dimensionality and the geometry of location: use... two-dimensional coordinate systems to represent... line segments, and figures. Objectives Find the magnitude and direction of a vector. Use vectors and vector addition to solve realworld problems. Who uses this? By using vectors, a kayaker can take water currents into account when planning a course. (See Example 5.) The speed and direction an object moves can be represented by a vector. A vector is a quantity that has both length and direction. You can
think of a vector as a directed line segment. The vector below may be named  AB or  v. Vocabulary vector component form magnitude direction equal vectors parallel vectors resultant vector Also G.1.B, G.7.C, G.11.C A vector can also be named using component form. The component form 〈x, y〉 of a vector lists the horizontal and vertical change from the initial point to   CD is 〈2, 3〉. the terminal point. The component form of E X A M P L E 1 Writing Vectors in Component Form Write each vector in component form. A B   EF The horizontal change from E to F is 4 units. The vertical change from E to F is -3 units. So the component form of  EF is 〈4, -3〉.   PQ with P (7, -5) and Q (4, 3)  PQ = 〈〉 Subtract the coordinates of the initial point  PQ = 〈4 - 7, 3 - (-5) 〉  PQ = 〈-3, 8〉 from the coordinates of the terminal point. Substitute the coordinates of the given points. Simplify. Write each vector in component form. 1a.  u 1b. the vector with initial point L (-1, 1) and terminal point M (6, 2) 8-6 Vectors 559 559 �� The magnitude of a vector is its length. The magnitude of a vector is written  AB ⎟ or ⎜ ⎜  v ⎟. When a vector is used to represent speed in a given direction, the magnitude of the vector equals the speed. For example, if a vector represents the course a kayaker paddles, the magnitude of the vector is the kayaker’s speed. E
X A M P L E 2 Finding the Magnitude of a Vector Draw the vector 〈4, -2〉 on a coordinate plane. Find its magnitude to the nearest tenth. Step 1 Draw the vector on a coordinate plane. Use the origin as the initial point. Then (4, -2) is the terminal point. Step 2 Find the magnitude. Use the Distance Formula. ⎜〈4, -2〉⎟ = √  (4 - 0) 2 + (-2 - 0) 2 = √  20 ≈ 4.5 2. Draw the vector 〈-3, 1〉 on a coordinate plane. Find its magnitude to the nearest tenth. The direction of a vector is the angle that it makes with a horizontal line. This angle is measured counterclockwise from the positive x-axis.  AB is 60°. The direction of See Lesson 4-5, page 252, to review bearings. The direction of a vector can also be given as a bearing relative to the compass directions north, south, east, and west. of N 30° E.  AB has a bearing E X A M P L E 3 Finding the Direction of a Vector A wind velocity is given by the vector 〈2, 5〉. Draw the vector on a coordinate plane. Find the direction of the vector to the nearest degree. Step 1 Draw the vector on a coordinate plane. Use the origin as the initial point. Step 2 Find the direction. Draw right triangle ABC as shown. ∠A is the angle formed by the vector and the x-axis, and tan A = 5 _ 2. So m∠A = tan -1 ( 5 _ ) ≈ 68°. 2 3. The force exerted by a tugboat is given by the vector 〈7, 3〉. Draw the vector on a coordinate plane. Find the direction of the vector to the nearest degree. 560 560 Chapter 8 Right Triangles and Trigonometry �� Two vectors are equal vectors if they have the same  v.  u = magnitude and the same direction. For example, Equal vectors do not have to have the same
initial point and terminal point. Two vectors are parallel vectors if they have the same direction or if they have opposite directions. They may have different magnitudes. For example, vectors are always parallel vectors.  x. Equal  w ǁ  BA  AB ≠ Note that since these vectors do not have the same direction. E X A M P L E 4 Identifying Equal and Parallel Vectors Identify each of the following. A equal vectors  AB =  GH Identify vectors with the same magnitude and direction. B parallel vectors  AB ǁ  GH and  CD ǁ  EF Identify vectors with the same or opposite directions. Identify each of the following. 4a. equal vectors 4b. parallel vectors The resultant vector is the vector that represents the sum of two given vectors. To add two vectors geometrically, you can use the head-to-tail method or the parallelogram method. Vector Addition METHOD EXAMPLE Head-to-Tail Method Place the initial point (tail) of the second vector on the terminal point (head) of the first vector. The resultant is the vector that joins the initial point of the first vector to the terminal point of the second vector. Parallelogram Method Use the same initial point for both of the given vectors. Create a parallelogram by adding a copy of each vector at the terminal point (head) of the other vector. The resultant vector is a diagonal of the parallelogram formed. 8-6 Vectors 561 561 �� To add vectors numerically, add their components. If  v = 〈 x 2, y 2 〉, then  〉.  〉 and E X A M P L
E 5 Sports Application A kayaker leaves shore at a bearing of N 55° E and paddles at a constant speed of 3 mi/h. There is a 1 mi/h current moving due east. What are the kayak’s actual speed and direction? Round the speed to the nearest tenth and the direction to the nearest degree. Step 1 Sketch vectors for the kayaker and the current. Component form gives the horizontal and vertical change from the initial point to the terminal point of the vector. Step 2 Write the vector for the kayaker in component form. The kayaker’s vector has a magnitude of 3 mi/h and makes an angle of 35° with the x-axis., so x = 3 cos 35° ≈ 2.5. cos 35° = x _ 3 y _, so y = 3 sin 35° ≈ 1.7. sin 35° = 3 The kayaker’s vector is 〈2.5, 1.7〉. Step 3 Write the vector for the current in component form. Since the current moves 1 mi/h in the direction of the x-axis, it has a horizontal component of 1 and a vertical component of 0. So its vector is 〈1, 0〉. Step 4 Find and sketch the resultant vector AB. Add the components of the kayaker’s vector and the current’s vector. 〈2.5, 1.7〉 + 〈1, 0〉 = 〈3.5, 1.7〉 The resultant vector in component form is 〈3.5, 1.7〉. Step 5 Find the magnitude and direction of the resultant vector. The magnitude of the resultant vector is the kayak’s actual speed. ⎜〈3.5, 1.7〉⎟ = √ (3.5 - 0)2 + (1.7 - 0)2 ≈ 3.9 mi/h The angle measure formed by the resultant vector gives the kayak’s actual direction. tan A = 1.7_ 3.5 3.5) ≈ 26°, or N 64° E., so A = tan -1(1.7
_ 5. What if…? Suppose the kayaker in Example 5 instead paddles at 4 mi/h at a bearing of N 20° E. What are the kayak’s actual speed and direction? Round the speed to the nearest tenth and the direction to the nearest degree. 562 562 Chapter 8 Right Triangles and Trigonometry ��〈��〉�� THINK AND DISCUSS 1. Explain why the segment with endpoints (0, 0) and (1, 4) is not a vector. 2. Assume you are given a vector in component form. Other than the Distance Formula, what theorem can you use to find the vector’s magnitude? 3. Describe how to add two vectors numerically. 4. GET ORGANIZED Copy and complete the graphic organizer. 8-6 Exercises Exercises KEYWORD: MG7 8-6 KEYWORD: MG7 Parent GUIDED PRACTICE Vocabulary Apply the vocabulary from this lesson to answer each question. 1. 2.? vectors have the same magnitude and direction. (equal, parallel, or resultant) ̶̶̶̶? vectors have the same or opposite directions. (equal, parallel, or resultant) ̶̶̶̶ 3. The? of a vector indicates the vector’s size. (magnitude or direction) ̶̶̶̶ Write each vector in component form. p. 559 4.  AC with A (1, 2) and C (6, 5) 5. the vector with initial point M (-4, 5) and terminal point N (4, -3) 6.  PQ Draw each vector on a coordinate plane. Find its magnitude to the nearest tenth. p. 560 7. 〈1, 4〉 8. 〈-3, -2〉 9. 〈5, -3. 560 Draw each vector on a coordinate plane. Find the direction of the vector to the nearest degree. 10. A river’s current is given by the vector 〈4, 6〉. 11. The velocity of a plane is given by the vector 〈5, 1〉. 12. The path of a hiker is given by the vector 〈6
, 3〉. Identify each of the following. p. 561 13. equal vectors in diagram 1 14. parallel vectors in diagram 1 15. equal vectors in diagram 2 16. parallel vectors in diagram 2 8-6 Vectors 563 563 ��. 562 17. Recreation To reach a campsite, a hiker first walks for 2 mi at a bearing of N 40° E. Then he walks 3 mi due east. What are the magnitude and direction of his hike from his starting point to the campsite? Round the distance to the nearest tenth of a mile and the direction to the nearest degree. PRACTICE AND PROBLEM SOLVING Independent Practice Write each vector in component form. For See Exercises Example 18. JK with J (-6, -7) and K (3, -5)  18–20 21–23 24–26 27–30 31 1 2 3 4 5 TEKS TEKS TAKS TAKS Skills Practice p. S19 Application Practice p. S35 19.  EF with E (1.5, -3) and F (-2, 2.5) 20.  w Draw each vector on a coordinate plane. Find its magnitude to the nearest tenth. 21. 〈-2, 0〉 22. 〈1.5, 1.5〉 23. 〈2.5, -3.5〉 Draw each vector on a coordinate plane. Find the direction of the vector to the nearest degree. 24. A boat’s velocity is given by the vector 〈4, 1.5〉. 25. The path of a submarine is given by the vector 〈3.5, 2.5〉. 26. The path of a projectile is given by the vector 〈2, 5〉. Identify each of the following. 27. equal vectors in diagram 1 28. parallel vectors in diagram 1 29. equal vectors in diagram 2 30. parallel vectors in diagram 2 31. Aviation The pilot of a single-engine airplane flies at a constant speed of 200 km/h at a bearing of N 25° E. There is a 40 km/h crosswind blowing southeast (S 45° E). What are the
plane’s actual speed and direction? Round the speed to the nearest tenth and the direction to the nearest degree. Find each vector sum. 32. 〈1, 2〉 + 〈0, 6〉 34. 〈0, 1〉 + 〈7, 0〉 33. 〈-3, 4〉 + 〈5, -2〉 35. 〈8, 3〉 + 〈-2, -1〉 36. Critical Thinking Is vector addition commutative? That is, is  u +  v equal to  v +  u? Use the head-to-tail method of vector addition to explain why or why not. 564 564 Chapter 8 Right Triangles and Trigonometry 40°2 mi3 miCampsiteNSWEge07se_c08L06002ahiking mapGeometry 2007 SEHolt Rinehart WinstonKaren Minot(415)883-6560�� 37. This problem will prepare you for the Multi-Step TAKS Prep on page 568. A helicopter at H must fly at 50 mi/h in the direction N 45° E to reach the site of a flood victim F. There is a 41 mi/h wind in the direction N 53° W.  HX he should use so that The pilot needs to the know the velocity vector  HF. his resultant vector will be a. What is m∠F? (Hint: Consider a vertical line through F.) b. Use the Law of Cosines to find the magnitude of  HX to the nearest tenth. c. Use the Law of Sines to find m∠FHX to the nearest degree. d. What is the direction of  HX? Write each vector in component form. Round values to the nearest tenth. 38. magnitude 15, direction 42° 39. magnitude 7.2, direction 9° 40. magnitude 12.1, direction N 57° E 41. magnitude 5.8, direction N 22° E 42. Physics A classroom has a window near the ceiling,
and a long pole must be used to close it. a. Carla holds the pole at a 45° angle to the floor and applies 10 lb of force to the upper edge of the window. Find the vertical component of the vector representing the force on the window. Round to the nearest tenth. b. Taneka also applies 10 lb of force to close the window, but she holds the pole at a 75° angle to the floor. Find the vertical component of the force vector in this case. Round to the nearest tenth. c. Who will have an easier time closing the window, Carla or Taneka? (Hint: Who applies more vertical force?) 43. Probability The numbers 1, 2, 3, and 4 are written on slips of paper and placed in a hat. Two different slips of paper are chosen at random to be the x- and y-components of a vector. a. What is the probability that the vector will be equal to 〈1, 2〉? b. What is the probability that the vector will be parallel to 〈1, 2〉? 44. Estimation Use the vector 〈4, 6〉 to complete the following. a. Draw the vector on a sheet of graph paper. b. Estimate the vector’s direction to the nearest degree. c. Use a protractor to measure the angle the vector makes with a horizontal line. d. Use the vector’s components to calculate its direction. e. How did your estimate in part b compare to your measurement in part c and your calculation in part d? Multi-Step Find the magnitude of each vector to the nearest tenth and the direction of each vector to the nearest degree. 45.  u 47.  w 46.  v 48.  z 8-6 Vectors 565 565 �� 49. Football Write two vectors in component form to represent the pass pattern that Jason is told to run. Find the resultant vector and show that Jason’s move is equivalent to the vector. For each given vector, find another vector that has the same magnitude but a different direction. Then find a vector that has the same direction but a different magnitude. 50. 〈-3, 6〉 51. 〈12, 5〉 52. 〈8, -11〉 Multi-Step Find the sum of each pair
of vectors. Then find the magnitude and direction of the resultant vector. Round the magnitude to the nearest tenth and the direction to the nearest degree. 53.  u = 〈1, 2〉,  v = 〈4.8, -3.1〉  v = 〈2.5, -1〉  u = 〈-2, 7〉, 54. 55.  u = 〈6, 0〉,  v = 〈-2, 4〉 56.  u = 〈-1.2, 8〉,  v = 〈5.2, -2.1〉 57. Math History In 1827, the mathematician August Ferdinand Möbius published a book in which he introduced directed line segments (what we now call vectors). He showed how to perform scalar multiplication of vectors. For example, consider a hiker who walks along a path given by the vector walks twice as far in the same direction is given by the vector 2 a. Write the component form of the vectors b. Find the magnitude of c. Find the direction of d. Given the component form of a vector, explain how to find  v. How do they compare?  v. How do they compare?  v. The path of another hiker who  v and 2  v and 2  v and 2  v.  v. the components of the vector k  v, where k is a constant. e. Use scalar multiplication with k = -1 to write the negation of a vector  v in component form. 58. Critical Thinking A vector Another vector possible directions and magnitudes for the resultant vector.  u points due west with a magnitude of u units.  v points due east with a magnitude of v units. Describe three Math History August Ferdinand Möbius is best known for experimenting with the Möbius strip, a three-dimensional figure that has only one side and one edge. 59. Write About It Compare a line segment, a ray, and a vector. 566 566 Chapter 8 Right Triangles and Trigonometry �� 60. Which vector is parallel to 〈2, 1〉? �
� u  v  w  z 61. The vector 〈7, 9〉 represents the velocity of a helicopter. What is the direction of this vector to the nearest degree? 38° 52° 128° 142° 62. A canoe sets out on a course given by the vector 〈5, 11〉. What is the length of the canoe’s course to the nearest unit? 6 8 12 16 63. Gridded Response  AB has an initial point of (-3, 6) and a terminal point of (-5, -2). Find the magnitude of  AB to the nearest tenth. CHALLENGE AND EXTEND Recall that the angle of a vector’s direction is measured counterclockwise from the positive x-axis. Find the direction of each vector to the nearest degree. 64. 〈-2, 3〉 65. 〈-4, 0〉 66. 〈-5, -3〉 67. Navigation The captain of a ship is planning to sail in an area where there is a 4 mi/h current moving due east. What speed and bearing should the captain maintain so that the ship’s actual course (taking the current into account) is 10 mi/h at a bearing of N 70° E? Round the speed to the nearest tenth and the direction to the nearest degree. 68. Aaron hikes from his home to a park by walking 3 km at a bearing of N 30° E, then 6 km due east, and then 4 km at a bearing of N 50° E. What are the magnitude and direction of the vector that represents the straight path from Aaron’s home to the park? Round the magnitude to the nearest tenth and the direction to the nearest degree. SPIRAL REVIEW Solve each system of equations by graphing. (Previous course) x - y = -5 ⎧ 69. ⎨ ⎩ y = 3x + 1 x - 2y = 0 ⎧ 70. ⎨ ⎩ 2y + x = 8 x + y = 5 ⎧ 71. ⎨ ⎩ 3y + 15 = 2x Given that △JLM ∼ △NPS, the perimeter of △JLM is 12 cm, and the area of �
�JLM is 6 cm 2, find each measure. (Lesson 7-5) 72. the perimeter of △NPS 73. the area of △NPS Find each measure. Round lengths to the nearest tenth and angle measures to the nearest degree. (Lesson 8-5) 74. BC 75. m∠B 76. m∠C 8-6 Vectors 567 567 �� SECTION 8B Applying Trigonometric Ratios Help Is on the Way! Rescue helicopters were first used in the 1950s during the Korean War. The helicopters made it possible to airlift wounded soldiers to medical stations. Today, helicopters are used to rescue injured hikers, flood victims, and people who are stranded at sea. 1. The pilot of a helicopter is searching for an injured hiker. While flying at an altitude of 1500 ft, the pilot sees smoke at an angle of depression of 14°. Assuming that the smoke is a distress signal from the hiker, what is the helicopter’s horizontal distance to the hiker? Round to the nearest foot. 2. The pilot plans to fly due north at 100 mi/h from the helicopter’s current position H to the location of the smoke S. However there is a 30 mi/h wind in the direction N 57° W. The pilot needs to know  HA that he should use so that the velocity vector his resultant vector will be then use the Law of Cosines to find the magnitude of  HA to the nearest mile per hour.  HS. Find m∠S and 3. Use the Law of Sines to find the direction of  HA to the nearest degree. 568 568 Chapter 8 Right Triangles and Trigonometry �� SECTION 8B Quiz for Lessons 8-4 Through 8-6 8-4 Angles of Elevation and Depression 1. An observer in a blimp sights a football stadium at an angle of depression of 34°. The blimp’s altitude is 1600 ft. What is the horizontal distance from the blimp to the stadium? Round to the nearest foot. 2. When the angle of elevation of the sun is 78°, a building casts a shadow that is
6 m long. What is the height of the building to the nearest tenth of a meter? 8-5 Law of Sines and Law of Cosines Find each measure. Round lengths to the nearest tenth and angle measures to the nearest degree. 3. m∠A 4. GH 5. XZ 6. UV 7. m∠F 8. QS 8-6 Vectors Draw each vector on a coordinate plane. Find its magnitude to the nearest tenth. 9. 〈3, 1〉 10. 〈-2, -4〉 11. 〈0, 5〉 Draw each vector on a coordinate plane. Find the direction of the vector to the nearest degree. 12. A wind velocity is given by the vector 〈2, 1〉. 13. The current of a river is given by the vector 〈5, 3〉. 14. The force of a spring is given by the vector 〈4, 4〉. 15. To reach an island, a ship leaves port and sails for 6 km at a bearing of N 32° E. It then sails due east for 8 km. What are the magnitude and direction of the voyage directly from the port to the island? Round the distance to the nearest tenth of a kilometer and the direction to the nearest degree. Ready to Go On? 569 569 �� EXTENSION Trigonometry and EXTENSION the Unit Circle TEKS G.11.A Similarity and the geometry of shape: use and extend... transformations to expore... geometric figures. Also G.5.B, G.5.C, G.7.A, G.10.A, G.11.C Objective Define trigonometric ratios for angle measures greater than or equal to 90°. Rotations are used to extend the concept of trigonometric ratios to angle measures greater than or equal to 90°. Consider a ray with its endpoint at the origin, pointing in the direction of the positive x-axis. Rotate the ray counterclockwise around the origin. The acute angle formed by the ray and the nearest part of the positive or negative x-axis is called the reference angle. The rotated ray is called the terminal side of that angle. Vocabulary reference angle unit circle Angle measure: 135° Reference angle: 45° Angle measure:
345° Reference angle: 15° Angle measure: 435° Reference angle: 75° E X A M P L E 1 Finding Reference Angles Sketch each angle on the coordinate plane. Find the measure of its reference angle. A 102° B 236° Reference angle: 180° - 102° = 78° Reference angle: 236° - 180° = 56° Sketch each angle on the coordinate plane. Find the measure of its reference angle. 1a. 309° 1b. 410° The unit circle is a circle with a radius of 1 unit, centered at the origin. It can be used to find the trigonometric ratios of an angle. Consider the acute angle θ. Let P (x, y) be the point where the terminal side of θ intersects the unit circle. Draw a vertical line from P to the x-axis. Since cos θ = x __ 1 y __ 1, the coordinates of P can be written and sin θ = as (cos θ, sin θ). Thus if you know the coordinates of a point on the unit circle, you can find the trigonometric ratios for the associated angle. In trigonometry, the Greek letter theta, θ, is often used to represent angle measures. 570 570 Chapter 8 Right Triangles and Trigonometry �� E X A M P L E 2 Finding Trigonometric Ratios Find each trigonometric ratio. A cos 150° Sketch the angle on the coordinate plane. The reference angle is 30°. cos 30° = √  3 _ 2 sin 30° = 1 _ 2 Be sure to use the correct sign when assigning coordinates to a point on the unit circle. Let P (x, y) be the point where the terminal side of the angle intersects the unit circle. Since P is in Quadrant II, its x-coordinate is negative, and its y-coordinate is positive. So the coordinates of P are (- √  3 ___ 2, 1 __ 2 ). The cosine of 150° is the x-coordinate of P, so cos 150° = - √  3 ___ 2. B tan 315° Sketch the angle on the coordinate plane. The reference angle is 45°. cos 45° = √  2 _ 2 sin 45° = √  2 _ 2 Since P (x,
y) is in Quadrant IV, its y-coordinate is negative. So the coordinates of P are ( √  2 ___ 2, - √  2 ___ 2 ). Remember that tan θ = sin θ _. So tan 315° = sin 315° _ = cos θ cos 315° - √  2 ___ 2 _ √  2 ___ 2 = -1. Find each trigonometric ratio. 2a. cos 240° 2b. sin 135° EXTENSION Exercises Exercises Sketch each angle on the coordinate plane. Find the measure of its reference angle. 1. 125° 2. 216° 3. 359° Find each trigonometric ratio. 4. cos 225° 7. tan 135° 10. sin 90° 5. sin 120° 8. cos 420° 11. cos 180° 6. cos 300° 9. tan 315° 12. sin 270° 13. Critical Thinking Given that cos θ = 0.5, what are the possible values for θ between 0° and 360°? 14. Write About It Explain how you can use the unit circle to find tan 180°. 15. Challenge If sin θ ≈ -0.891, what are two values of θ between 0° and 360°? Chapter 8 Extension 571 571 �� For a complete list of the postulates and theorems in this chapter, see p. S82. Vocabulary angle of depression......... 544 equal vectors............... 561 sine........................ 525 angle of elevation........... 544 geometric mean............ 519 tangent.................... 525 component form........... 559 magnitude................. 560 trigonometric ratio......... 525 cosine..................... 525 parallel vectors......
....... 561 vector...................... 559 direction................... 560 resultant vector............. 561 Complete the sentences below with vocabulary words from the list above. 1. The? of a vector gives the horizontal and vertical change from the initial point ̶̶̶̶ to the terminal point. 2. Two vectors with the same magnitude and direction are called 3. If a and b are positive numbers, then √  ab is the?. ̶̶̶̶? of a and b. ̶̶̶̶ 4. A(n)? is the angle formed by a horizontal line and a line of sight to a point ̶̶̶̶ above the horizontal line. 5. The sine, cosine, and tangent are all examples of a(n)?. ̶̶̶̶ 8-1 Similarity in Right Triangles (pp. 518–523) E X A M P L E S EXERCISES ■ Find the geometric mean of 5 and 30. 6. Write a similarity Let x be the geometric mean. x 2 = (5)(30) = 150 Def. of geometric mean x = √150 = 5 √6 Find the positive square root. statement comparing the three triangles. TEKS G.5.B, G.5.D, G.8.A, G.8.C, G.11.A, G.11.B, G.11.C Find the geometric mean of each pair of numbers. If necessary, give the answer in simplest radical form. 7. 1 _ 4 8. 3 and 17 and 100 √  33 is the geometric mean of 3 and 3 + x. Find x, y, and z. 9. 10. ■ Find x, y, and z. ( √  33 ) 2 = 3 (3 + x) 33 = 9 + 3x 24 = 3x x = 8 y 2 = (3) (8) y 2 = 24 y = √  24 = 2 √  6 y is the geometric mean of 3 and 8. 11. z 2 = (
8) (11) z 2 = 88 z = √  88 = 2 √  22 z is the geometric mean of 8 and 11. 572 572 Chapter 8 Right Triangles and Trigonometry �� 8-2 Trigonometric Ratios (pp. 525–532) TEKS G.5.B, G.5.D, G.8.C, G.11.B, G.11.C E X A M P L E S EXERCISES Find each length. Round to the nearest hundredth. Find each length. Round to the nearest hundredth. 12. UV ■ EF sin 75° = EF _ 8.1 EF = 8.1 (sin 75°) EF ≈ 7.82 cm ■ AB tan 34° = 4.2 _ AB AB tan 34° = 4.2 AB = 4.2 _ tan 34° AB ≈ 6.23 in. Since the opp. leg and hyp. are involved, use a sine ratio. 13. PR 14. XY 15. JL Since the opp. and adj. legs are involved, use a tangent ratio. 8-3 Solving Right Triangles (pp. 534–541) TEKS G.5.D, G.7.A, G.7.C, G.8.C, G.11.B, G.11.C E X A M P L E EXERCISES ■ Find the unknown measures in △LMN. Round lengths to the nearest hundredth and angle measures to the nearest degree. Find the unknown measures. Round lengths to the nearest hundredth and angle measures to the nearest degree. 16. The acute angles of a right triangle are 17. complementary. So m∠N = 90° - 61° = 29°. sin L = MN _ Write a trig. ratio. LN sin 61° = 8.5 _ LN LN = 8.5 _ sin 61° tan L = MN _ LM tan 61° = 8.5 _ LM LM = 8.5 _ tan 61° ≈ 9.72 ≈ 4.71 Substitute the given 18. 19. values. Solve for LN. Write a trig. ratio. Substitute the given values. Solve for LM. Study Guide: Review 573 573 ��
�� 8-4 Angles of Elevation and Depression (pp. 544–549) TEKS G.5.D, G.11.C E X A M P L E S EXERCISES Classify each angle as an angle of elevation or angle of depression. ■ A pilot in a plane spots a forest fire on the ground at an angle of depression of 71°. The plane’s altitude is 3000 ft. What is the horizontal distance from the plane to the fire? Round to the nearest foot. tan 71° = 3000_ XF XF = 3000_ tan 71° XF ≈ 1033 ft ■ A diver is swimming at a depth of 63 ft below sea level. He sees a buoy floating at sea level at an angle of elevation of 47°. How far must the diver swim so that he is directly beneath the buoy? Round to the nearest foot. tan 47° = 63 _ XD XD = 63 _ tan 47° XD ≈ 59 ft 20. ∠1 21. ∠2 22. When the angle of elevation to the sun is 82°, a monument casts a shadow that is 5.1 ft long. What is the height of the monument to the nearest foot? 23. A ranger in a lookout tower spots a fire in the distance. The angle of depression to the fire is 4°, and the lookout tower is 32 m tall. What is the horizontal distance to the fire? Round to the nearest meter. 8-5 Law of Sines and Law of Cosines (pp. 551–558) TEKS G.5.B, G.5.D, G.7.A, G.11.A, G.11.C E X A M P L E S EXERCISES Find each measure. Round lengths to the nearest tenth and angle measures to the nearest degree. ■ m∠B Find each measure. Round lengths to the nearest tenth and angle measures to the nearest degree. 24. m∠Z sin B _ AC sin B _ 6 = sin C _ AB = sin 88° _ 8 Law of Sines Substitute the given values. 25. MN sin B = 6 sin 88° _ m∠B = sin -1 ( 6 sin 88° _ 8 8 ) ≈ 49° Multiply both sides by 6. 574 574 Chapter 8 Right Triangles and Trigonometry ��
�� Find each measure. Round lengths to the nearest tenth and angle measures to the nearest degree. ■ HJ Find each measure. Round lengths to the nearest tenth and angle measures to the nearest degree. 26. EF Use the Law of Cosines. 27. m∠Q HJ 2 = GH 2 + G J 2 - 2 (GH) (GJ) cos G =10 ) (11 ) cos 32° H J 2 ≈ 34.4294 HJ ≈ 5.9 Simplify. Find the square root. 8-6 Vectors (pp. 559–567) TEKS G.1.B, G.7.A, G.7.C, G.11.C E X A M P L E S EXERCISES ■ Draw the vector 〈-1, 4〉 on a coordinate plane. Find its magnitude to the nearest tenth. ⎜〈-1, 4〉⎟ = √ (-1)2 + (4)2 Write each vector in component form. 28.  AB with A (5, 1) and B (-2, 3) 29.  MN with M (-2, 4) and N (-1, -2) = √17 ≈ 4.1 30.  RS ■ The velocity of a jet is given by the vector 〈4, 3〉. Draw the vector on a coordinate plane. Find the direction of the vector to the nearest degree. In △PQR, tan P = 3 _ 4, so m∠P = tan -1 ( 3 _ ) ≈ 37°. 4 ■ Susan swims across a river at a bearing of N 75° E at a speed of 0.5 mi/h. The river’s current moves due east at 1 mi/h. Find Susan’s actual speed to the nearest tenth and her direction to the nearest degree. cos 15° = x _ 0.5 y _ 0.5 sin 15° =, so x ≈ 0.48., so y ≈ 0.13. Susan’s vector is
〈0.48, 0.13〉. The current is 〈1, 0〉. Susan’s actual speed is the magnitude of the resultant vector, 〈1.48, 0.13〉. ⎜〈1.48, 0.13〉⎟ = Her direction is tan -1 ( 0.13 _ √  (1.48) 2 + (0.13) 2 ≈ 1.5 mi/h ) ≈ 5°, or N 85° E. 1.48 Draw each vector on a coordinate plane. Find its magnitude to the nearest tenth. 31. 〈-5, -3〉 32. 〈-2, 0〉 33. 〈4, -4〉 Draw each vector on a coordinate plane. Find the direction of the vector to the nearest degree. 34. The velocity of a helicopter is given by the vector 〈4, 5〉. 35. The force applied by a tugboat is given by the vector 〈7, 2〉. 36. A plane flies at a constant speed of 600 mi/h at a bearing of N 55° E. There is a 50 mi/h crosswind blowing due east. What are the plane’s actual speed and direction? Round the speed to the nearest tenth and the direction to the nearest degree. Study Guide: Review 575 575 �� Find x, y, and z. 1. 2. 3. Use a special right triangle to write each trigonometric ratio as a fraction. 4. cos 60° 5. sin 45° 6. tan 60° Find each length. Round to the nearest hundredth. 7. PR 8. AB 9. FG 10. Nate built a skateboard ramp that covers a horizontal distance of 10 ft. The ramp rises a total of 3.5 ft. What angle does the ramp make with the ground? Round to the nearest degree. 11. An observer at the top of a skyscraper sights a tour bus at an angle of depression of 61°. The skyscraper is 910 ft tall. What is the horizontal distance from the base of the skyscraper to the tour bus? Round to the nearest foot. Find each measure
. Round lengths to the nearest tenth and angle measures to the nearest degree. 12. m∠B 13. RS 14. m∠M Draw each vector on a coordinate plane. Find its magnitude to the nearest tenth. 15. 〈1, 3〉 16. 〈-4, 1〉 17. 〈2, -3〉 Draw each vector on a coordinate plane. Find the direction of the vector to the nearest degree. 18. The velocity of a plane is given by the vector 〈3, 5〉. 19. A wind velocity is given by the vector 〈4, 1〉. 20. Kate is rowing across a river. She sets out at a bearing of N 40° E and paddles at a constant rate of 3.5 mi/h. There is a 2 mi/h current moving due east. What are Kate’s actual speed and direction? Round the speed to the nearest tenth and the direction to the nearest degree. 576 576 Chapter 8 Right Triangles and Trigonometry �� FOCUS ON SAT MATHEMATICS SUBJECT TESTS Though you can use a calculator on the SAT Mathematics Subject Tests, it may be faster to answer some questions without one. Remember to use test-taking strategies before you press buttons! 4. A swimmer jumps into a river and starts swimming directly across it at a constant velocity of 2 meters per second. The speed of the current is 7 meters per second. Given the current, what is the actual speed of the swimmer to the nearest tenth? (A) 0.3 meters per second (B) 1.7 meters per second (C) 5.0 meters per second (D) 7.3 meters per second (E) 9.0 meters per second 5. What is the approximate measure of the vertex angle of the isosceles triangle below? (A) 28.1° (B) 56.1° (C) 62.0° (D) 112.2° (E) 123.9° The SAT Mathematics Subject Tests each consist of 50 multiple-choice questions. You are not expected to have studied every topic on the SAT Mathematics Subject Tests, so some questions may be unfamiliar. You may want to time yourself as you take this practice test. It should take you
about 6 minutes to complete. 1. Let P be the acute angle formed by the line -x + 4y = 12 and the x-axis. What is the approximate measure of ∠P? (A) 14° (B) 18° (C) 72° (D) 76° (E) 85° 2. In right triangle DEF, DE = 15, EF = 36, and DF = 39. What is the cosine of ∠F? (A) 5 _ 12 (B) 12 _ 5 (C) 5 _ 13 (D) 12 _ 13 (E) 13 _ 12 3. A triangle has angle measures of 19°, 61°, and 100°. What is the approximate length of the side opposite the 100° angle if the side opposite the 61° angle is 8 centimeters long? (A) 2.5 centimeters (B) 3 centimeters (C) 9 centimeters (D) 12 centimeters (E) 13 centimeters College Entrance Exam Practice 577 577 �� Any Question Type: Estimate Once you find the answer to a test problem, take a few moments to check your answer by using estimation strategies. By doing so, you can verify that your final answer is reasonable. Gridded Response Find the geometric mean of 38 and 12 to the nearest hundredth. Let x be the geometric mean. x 2 = (38) (12) = 456 Def. of geometric mean x ≈ 21.35 Find the positive square root. Now use estimation to check that this answer is reasonable. x 2 ≈ (40) (10) = 400 Round 38 to 40 and round 12 to 10. x ≈ 20 Find the positive square root. The estimate is close to the calculated answer, so 21.35 is a reasonable answer. Multiple Choice Which of the following is equal to sin X? 0.02 0.41 0.91 2.44 Use a trigonometric ratio to find the answer. sin X = YZ _ XZ sin X = 9 _ 22 ≈ 0.41 The sine of an ∠ is opp. leg _ hyp.. Substitute the given values and simplify. Now use estimation to check that this answer is reasonable. sin X ≈ 10 _ 20 ≈ 0.5 Round 9 to 10 and round 22 to 20. The estimate is close to the calculated answer, so B is a reasonable answer. 578 578 Chapter 8 Right Triangles and Trigonometry �� An
extra minute spent checking your answers can result in a better test score. Item C Multiple Choice In △QRS, what is the ̶̶ SQ to the nearest tenth of a measure of centimeter? Read each test item and answer the questions that follow. Item A Gridded Response A cell phone tower casts a shadow that is 121 ft long when the angle of elevation to the sun is 48°. How tall is the cell phone tower? Round to the nearest foot. 1. A student estimated that the answer should be slightly greater than 121 by comparing tan 48° and tan 45°. Explain why this estimation strategy works. 2. Describe how to use the inverse tangent function to estimate whether an answer of 134 ft makes sense. Item B   BC has an initial point of Multiple Choice (-1, 0) and a terminal point of (4, 2). What are the magnitude and direction of   BC? 5.39; 22° 5.39; 68° 6.39; 22° 6.39; 68° 3. A student correctly found the magnitude of  BC as √  29. The student then calculated the value of this radical as 6.39. Explain how to use perfect squares to estimate the value of √  29. Is 6.39 a reasonable answer? 4. A student calculated the measure of the angle the vector forms with a horizontal line as 68°. Use estimation to explain why this answer is not reasonable. 9.3 centimeters 10.5 centimeters 30.1 centimeters 61.7 centimeters 5. A student calculated the answer as 30.1 cm. The student then used the diagram to estimate that SQ is more than half of RQ. So the student decided that his answer was reasonable. Is this estimation method a good way to check your answer? Why or why not? 6. Describe how to use estimation and the Pythagorean Theorem to check your answer to this problem. Item D Multiple Choice The McCleods have a variable interest rate on their mortgage. The rate is 2.625% the first year and 4% the following year. The average interest rate is the geometric mean of these two rates. To the nearest hundredth of a percent, what is the average interest rate for their mortgage? 1.38% 3.
24% 3.89% 10.50% 7. Describe how to use estimation to show that choices F and J are unreasonable. 8. To find the answer, a student uses the equation x 2 = (2.625) (4). Which compatible numbers should the student use to quickly check the answer? TAKS Tackler 579 579 �� KEYWORD: MG7 TestPrep CUMULATIVE ASSESSMENT, CHAPTERS 1–8 Multiple Choice 1. What is the length of ̶̶ UX to the nearest centimeter? 6. △ABC has vertices A (-2, -2), B (-3, 2), and C (1, 3). Which translation produces an image with vertices at the coordinates (-2, -2), (2, -1), and (-1, -6)? (x, y) → (x + 1, y - 4) (x, y) → (x + 2, y - 8) (x, y) → (x - 3, y - 5) (x, y) → (x - 4, y + 1) 7. △ABC is a right triangle in which m∠A = 30° and m∠B = 60°. Which of the following are possible lengths for the sides of this triangle? AB = √  3, AC = √  2, and BC = 1 AB = 4, AC = 2, and BC = 2 √  3 AB = 6 √  3, AC = 27, and BC = 3 √  3 AB = 8, AC = 4 √  3, and BC = 4 8. Based on the figure below, which of the following similarity statements must be true? △PQR ∼ △TSR △PQR ∼ △RTQ △PQR ∼ △TSQ △PQR ∼ △TQP 9. ABCD is a rhombus with vertices A (1, 1) and C (3, 4). Which of the following lines is parallel to diagonal ̶̶ BD? 2x - 3y = 12 2x + 3y = 12 3x + 2y = 12 3x - 4y = 12 3 centimeters 7 centimeters 9 centimeters 13 centimeters 2. △ABC is a
right triangle. m∠A = 20°, m∠B = 90°, AC = 8, and AB = 3. Which expression can be used to find BC? 3 _ tan 70° 8 _ sin 20° 8 tan 20° 3 cos 70° 3. A slide at a park is 25 ft long, and the top of the slide is 10 ft above the ground. What is the approximate measure of the angle the slide makes with the ground? 21.8° 23.6° 66.4° 68.2° 4. Which of the following vectors is equal to the vector with an initial point at (2, -1) and a terminal point at (-2, 4)? 〈-4, -5〉〈-4, 5〉〈5, -4〉〈5, 4〉 5. Which statement is true by the Addition Property of Equality? If 3x + 6 = 9y, then x + 2 = 3y. If t = 1 and s = t + 5, then s = 6. If k + 1 = ℓ + 2, then 2k + 2 = 2ℓ + 4. If a + 2 = 3b, then a + 5 = 3b + 3. 580 580 Chapter 8 Right Triangles and Trigonometry �� 10. Which of the following is NOT equivalent to sin 60°? cos 30° √  3 _ 2 (cos 60°) (tan 60°) tan 30° _ sin 30° 11. ABCDE is a convex pentagon. ∠A ≅ ∠B ≅ ∠C, ∠D ≅ ∠E, and m∠A = 2m∠D. What is the measure of ∠C? 67.5° 135° 154.2° 225° 12. Which of the following sets of lengths can represent the side lengths of an obtuse triangle? 4, 7.5, and 8.5 7, 12, and 13 9.5, 16.5, and 35 36, 75, and 88 �� Be sure to correctly identify any pairs of parallel lines before using the Alternate Interior Angles Theorem or the Same-Side Interior Angles Theorem. 13. What is the value of x? 22.5 45 90 135 Gridded Response 14. Find the next item in

EB = AF ___. FC Construction Triangle Proportionality Theorem Construct a line parallel to a side of a triangle.    Use a straightedge to draw △ABC. Label E on AB. Construct ∠E ≅ ∠B. Label the ̶̶ AC as F. intersection of   EF and ̶̶   EF ǁ BC by the Converse of the Corresponding Angles Postulate. 7- 4 Applying Properties of Similar Triangles 481 481 �� E X A M P L E 1 Finding the Length of a Segment Find CY. It is given that ̶̶ XY ǁ ̶̶ BC, so AX ___ XB = AY ___ YC by the Triangle Proportionality Theorem. = 10 _ 9 _ 4 CY 9 (CY ) = 40 CY = 40 _, or 4 4 _ 9 9 1. Find PN. Substitute 9 for AX, 4 for XB, and 10 for AY. Cross Products Prop. Divide both sides by 9. Theorem 7-4-2 Converse of the Triangle Proportionality Theorem THEOREM HYPOTHESIS CONCLUSION If a line divides two sides of a triangle proportionally, then it is parallel to the third side. AE _ EB = AF _ FC   EF ǁ ̶̶ BC You will prove Theorem 7-4-2 in Exercise 23. E X A M P L E 2 Verifying Segments are Parallel ̶̶̶ MN ǁ ̶̶ KL. = 2 Verify that = 42 _ 21 = 30 _ 15 = JN ___, NL JM _ MK JN _ NL Since JM ___ MK Triangle Proportionality Theorem. ̶̶̶ MN ǁ = 2 ̶̶ KL by the Converse of the 2. AC = 36 cm, and BC = 27 cm. ̶̶ DE ǁ Verify that ̶̶ AB. Corollary 7-4-3 Two-Transversal Proportionality THEOREM HYPOTHESIS CONCLUSION If three or more parallel lines intersect two transversals, then they divide the transversals proportionally. AC _ CE = BD _ DF You will prove Corollary 7-4-3 in Exercise 24. 482 48

2 Chapter 7 Similarity �� E X A M P L E 3 Art Application ̶̶ AK ǁ An artist used perspective to draw guidelines to help her sketch a row of parallel trees. She then checked the drawing by measuring the distances between the trees. What is LN? ̶̶̶ ̶̶ CM ǁ BL ǁ KL _ = AB _ LN BD BD = BC + CD BD = 1.4 + 2.2 = 3.6 cm 2.6 _ LN = 2.4 _ 3.6 ̶̶̶ DN Given 2-Transv. Proportionality Corollary Seg. Add. Post. Substitute 1.4 for BC and 2.2 for CD. Substitute the given values. 2.4 (LN) = 3.6 (2.6) LN = 3.9 cm Cross Products Prop. Divide both sides by 2.4. 3. Use the diagram to find LM and MN to the nearest tenth. The previous theorems and corollary lead to the following conclusion. Theorem 7-4-4 Triangle Angle Bisector Theorem THEOREM HYPOTHESIS CONCLUSION An angle bisector of a triangle divides the opposite side into two segments whose lengths are proportional to the lengths of the other two sides. (△ ∠ Bisector Thm.) BD _ DC = AB _ AC You will prove Theorem 7-4-4 in Exercise 38. E X A M P L E 4 Using the Triangle Angle Bisector Theorem Find RV and VT. RV ___ VT = SR ___ ST x + 2 _ 2x + 1 = 10 _ 14 by the △ ∠ Bisector Thm. Substitute the given values. You can check your answer by substituting the values into the proportion. = SR __ RV ___ VT ST = 10 __ 5 __ 14 7 = 5 __ 5 __ 7 7 14 (x + 2) = 10 (2x + 1) 14x + 28 = 20x + 10 18 = 6x x = 3 Cross Products Prop. Dist. Prop. Simplify. Divide both sides by 6. RV = x + 2 = 3 + 2 = 5 VT = 2x + 1 Substitute 3 for x. = 2 (3) + 1 = 7 4. Find AC and DC. 7- 4 Applying Properties of Similar Triangles 483 483 ��

�� THINK AND DISCUSS 1. ̶̶ BC. Use what you know about similarity ̶̶ XY ǁ and proportionality to state as many different proportions as possible. 2. GET ORGANIZED Copy and complete the graphic organizer. Draw a figure for each proportionality theorem or corollary and then measure it. Use your measurements to write an if-then statement about each figure. 7-4 Exercises Exercises GUIDED PRACTICE. 482. 482 Find the length of each segment. ̶̶̶ DG 1. Verify that the given segments are parallel. ̶̶ AB and ̶̶ CD 3. KEYWORD: MG7 7-4 KEYWORD: MG7 Parent ̶̶ RN 2. ̶̶ TU and ̶̶ RS 4. Travel The map shows the area around p. 483 Herald Square in Manhattan, New York, and the approximate length of several streets. If the numbered streets are parallel, what is the length of Broadway between 34th St. and 35th St. to the nearest foot? 484 484 Chapter 7 Similarity �� Find the length of each segment. p. 483 6. ̶̶ QR and ̶̶ RS ̶̶ CD and ̶̶ AD 7. PRACTICE AND PROBLEM SOLVING Find the length of each segment. ̶̶ KL 8. ̶̶ XZ 9. Independent Practice For See Exercises Example 8–9 10–11 12 13–14 1 2 3 4 TEKS TEKS TAKS TAKS Verify that the given segments are parallel. Skills Practice p. S17 Application Practice p. S34 10. ̶̶ AB and ̶̶ CD 11. ̶̶̶ MN and ̶̶ QR � �� 12. Architecture The wooden treehouse has horizontal siding that is parallel to the

base. What are LM and MN to the nearest hundredth? �� Find the length of each segment. 13. ̶̶ BC and ̶̶ CD 14. ̶̶ ST and ̶̶ TU = AC_ In the figure,    BC ǁ    DE ǁ    FG. Complete each proportion. 15. AB_ BD 17. DF _ = EG _ CE = AE_ EG = 16. _ DF 18. AF _ AB 20. AB_ AC _ AC = BF_ 19. BD_ CE = _ EG 21. The bisector of an angle of a triangle divides the opposite side of the triangle into segments that are 12 in. and 16 in. long. Another side of the triangle is 20 in. long. What are two possible lengths for the third side? 7- 4 Applying Properties of Similar Triangles 485 485 �� 22. This problem will prepare you for the Multi-Step TAKS Prep on page 502. Jaclyn is building a slide rail, the narrow, slanted beam found in skateboard parks. a. Write a proportion that Jaclyn can use to calculate the length of ̶̶ CE. b. Find CE. c. What is the overall length of the slide rail AJ? 23. Prove the Converse of the Triangle Proportionality Theorem. Given: AE _ EB Prove:   EF ǁ = AF _ FC ̶̶ BC 24. Prove the Two-Transversal Proportionality Corollary. Given:   AB ǁ   CD,   CD ǁ   EF Prove: AC _ = BD _ DF CE (Hint : Draw   BE through X.) 25. Given that   PQ ǁ   RS ǁ   TU a. Find

PR, RT, QS, and SU. b. Use your results from part b to write a proportion relating the segment lengths. Find the length of each segment. 26. ̶̶ EF 27. ̶̶ ST 28. Real Estate A developer is laying out lots along Grant Rd. whose total width is 500 ft. Given the width of each lot along Chavez St., what is the width of each of the lots along Grant Rd. to the nearest foot? 29. Critical Thinking Explain how to use a sheet of lined notebook paper to divide a segment into five congruent segments. Which theorem or corollary do you use? ̶̶ ̶̶ XY ǁ DE ǁ ̶̶ BC, ̶̶ AD 30. Given that Find EC. 31. Write About It In △ABC,  AD bisects ∠BAC. Write a proportionality statement for the triangle. What theorem supports your conclusion? 486 486 Chapter 7 Similarity �� 32. Which dimensions let you conclude that SR = 12, TR = 9 SR = 16, TR = 20 ̶̶ UV ǁ ̶̶ ST? SR = 35, TR = 28 SR = 50, TR = 48 33. In △ABC, the bisector of ∠A divides ̶̶ BC into segments with lengths 16 and 20. AC = 25. Which of these could be the length of ̶̶ AB? 20 12.8 16 18.75 34. On the map, 1st St. and 2nd St. are parallel. What is the distance from City Hall to 2nd St. along Cedar Rd.? 1.8 mi 3.2 mi 4.2 mi 5.6 mi 35. Extended Response Two segments are divided proportionally. The first segment is divided into lengths 20, 15, and x. The corresponding lengths in the second segment are 16, y, and 24. Find the value of x and y. Use these values and write six proportions. CHALLENGE AND EXTEND 36. The perimeter of △ABC is 29 m. ̶̶ AD bisects ∠A. Find AB and AC. 37. Prove that if two triangles are similar, then the ratio of their corresponding

angle bisectors is the same as the ratio of their corresponding sides. 38. Prove the Triangle Angle Bisector Theorem. ̶̶ AD bisects ∠A. Given: In △ABC, = AB _ Prove: BD _ AC DC ̶̶ BX ǁ ̶̶ AD and extend ̶̶ AC to X. Use properties Plan: Draw of parallel lines and the Converse of the Isosceles Triangle Theorem to show that Then apply the Triangle Proportionality Theorem. ̶̶ AX ≅ ̶̶ AB. 39. Construction Draw of similarity to divide ̶̶ AB any length. Use parallel lines and the properties ̶̶ AB into three congruent parts. SPIRAL REVIEW Write an algebraic expression that can be used to find the nth term of each sequence. (Previous course) 40. 5, 6, 7, 8,… 41. 3, 6, 9, 12,… 42. 1, 4, 9, 16,… 43. B is the midpoint of ̶̶ AC. A has coordinates (1, 4), and B has coordinates (3, -7). Find the coordinates of C. (Lesson 1-6) Verify that the given triangles are similar. (Lesson 7-3) 44. △ABC and △ADE 45. △JKL and △MLN 7- 4 Applying Properties of Similar Triangles 487 487 2.4 mi2.1 mi2.8 miCedar Rd.Aspen Rd.1st St.2nd St.CityHallLibraryge07se_c07l04007a�� 7-5 Using Proportional Relationships TEKS G.11.D Similarity and the geometry of shape: describe the effect on perimeter, area... when... dimensions of a figure are changed.... Also G.1.B, G.5.A, G.11.A, G.11.B Objectives Use ratios to make indirect measurements. Use scale drawings to solve problems. Vocabulary indirect measurement scale drawing scale Why learn this? Proportional relationships help you find distances that cannot be measured directly. Indirect measurement is any method that uses formulas, similar figures, and/or proportions to measure an object. The following example shows one indirect measurement technique. E X A M P L E 1 Measurement

Application Eiffel Tower replica in Paris, Texas A student wanted to find the height of a statue of a pineapple in Nambour, Australia. She measured the pineapple’s shadow and her own shadow. The student’s height is 5 ft 4 in. What is the height of the pineapple? Step 1 Convert the measurements to inches. AC = 5 ft 4 in. = (5 ⋅ 12) in. + 4 in. = 64 in. BC = 2 ft = (2 ⋅ 12) in. = 24 in. EF = 8 ft 9 in. = (8 ⋅ 12) in. + 9 in. = 105 in. Step 2 Find similar triangles. Because the sun’s rays are parallel, ∠1 ≅ ∠2. Therefore △ABC ∼ △DEF by AA ∼. Step 3 Find DF. = BC_ AC_ EF DF 64 _ = 24 _ 105 DF 24 (DF) = 64 ⋅ 105 DF = 280 Corr. sides are proportional. Substitute 64 for AC, 24 for BC, and 105 for EF. Cross Products Prop. Divide both sides by 24. The height of the pineapple is 280 in., or 23 ft 4 in. 1. A student who is 5 ft 6 in. tall measured shadows to find the height LM of a flagpole. What is LM? Whenever dimensions are given in both feet and inches, you must convert them to either feet or inches before doing any calculations. 488 488 Chapter 7 Similarity 8 ft 9 in.DFEge07se_ c07105002aaAB22 ftAB1C�� A scale drawing represents an object as smaller than or larger than its actual size. The drawing’s scale is the ratio of any length in the drawing to the corresponding actual length. For example, on a map with a scale of 1 cm : 1500 m, one centimeter on the map represents 1500 m in actual distance. E X A M P L E 2 Solving for a Dimension A proportion may compare measurements that have different units. The scale of this map of downtown Dallas is 1.5 cm : 300 m. Find the actual distance between Union Station and the Dallas Public Library. Use a ruler to measure the distance between Union Station and the Dallas Public Library. The distance is 6 cm. To find the actual distance x write a proportion comparing the map distance to the actual distance. 6 _ x = 1.

5 _ 300 1.5x = 6 (300) 1.5x = 1800 x = 1200 Cross Products Prop. Simplify. Divide both sides by 1.5. The actual distance is 1200 m, or 1.2 km. 2. Find the actual distance between City Hall and El Centro College. E X A M P L E 3 Making a Scale Drawing The Lincoln Memorial in Washington, D.C., is approximately 57 m long and 36 m wide. Make a scale drawing of the base of the building using a scale of 1 cm : 15 m. Step 1 Set up proportions to find the length ℓ and width w of the scale drawing. = 1 _ 15 15w = 36 ℓ _ = 1 _ 57 15 15ℓ = 57 w _ 36 ℓ = 3.8 m w = 2.4 cm Step 2 Use a ruler to draw a rectangle with these dimensions. 3. The rectangular central chamber of the Lincoln Memorial is 74 ft long and 60 ft wide. Make a scale drawing of the floor of the chamber using a scale of 1 in. : 20 ft. 7- 5 Using Proportional Relationships 489 489 S. LamarElmS. MarketS. HoustonS. GriffinFieldS. AustinCantonYoungWoodS. ErvayS. Akard JacksonCommerceMainCity HallDallas PublicLibraryUnion StationEl CentroCollege0300 mScale30Holt, Rinehart & WinstonGeometry © 2007ge07sec07105003a Dallas Area Street map2nd proof�� Similar Triangles Similarity, Perimeter, and Area Ratios STATEMENT RATIO △ABC ∼ △DEF = BC _ EF Perimeter ratio: Similarity ratio: AB _ = AC _ DE DF perimeter △ABC __ perimeter △DEF = 6 _ = 1 _ 4 24 Area ratio: area △ABC __ = 1 _ 2 = 12 _ 24 = ( 1 _ ) area △DEF 2 2 = 1 _ 2 The comparison of the similarity ratio and the ratio of perimeters and areas of similar triangles leads to the following theorem. Theorem 7-5-1 Proportional Perimeters and Areas Theorem If the similarity ratio of two similar figures is a __, then the ratio of their perimeters b, and the ratio of their areas is a 2 __ is a __ b 2 b, or ( a __ ) b. 2 You will prove Theorem 7-5-1 in

Exercises 44 and 45. E X A M P L E 4 Using Ratios to Find Perimeters and Areas Given that △RST ∼ △UVW, find the perimeter P and area A of △UVW. The similarity ratio of △RST to △UVW is 16 __ 20, or 4 __ 5. By the Proportional Perimeters and Areas Theorem, the ratio of the triangles’ perimeters is also 4 __ 5, and the ratio of the triangles’ areas is ( 4 __ 5 ), or 16 __ 25. 2 Perimeter 36 _ = 4 _ 5 P 4P = 5 (36) P = 45 ft Area = 16 _ 48 _ 25 A 16A = 25 ⋅ 48 A = 75 ft 2 The perimeter of △UVW is 45 ft, and the area is 75 ft 2. 4. △ABC ∼ △DEF, BC = 4 mm, and EF = 12 mm. If P = 42 mm and A = 96 mm 2 for △DEF, find the perimeter and area of △ABC. THINK AND DISCUSS 1. Explain how to find the actual distance between two cities 5.5 in. apart on a map that has a scale of 1 in. : 25 mi. 2. GET ORGANIZED Copy and complete the graphic organizer. Draw and measure two similar figures. Then write their ratios. 490 490 Chapter 7 Similarity �� 7-5 Exercises Exercises KEYWORD: MG7 7-5 KEYWORD: MG7 Parent GUIDED PRACTICE 1. Vocabulary Finding distances using similar triangles is called?. ̶̶̶̶ (indirect measurement or scale drawing ). Measurement To find the height of a dinosaur in p. 488 a museum, Amir placed a mirror on the ground 40 ft from its base. Then he stepped back 4 ft so that he could see the top of the dinosaur in the mirror. Amir’s eyes were approximately 5 ft 6 in. above the ground. What is the height of the dinosaur. 489 The scale of this blueprint of an art gallery is 1 in. : 48 ft. Find the actual lengths of the following walls. ̶̶ AB ̶̶ EF 3. 5. ̶̶ CD ̶̶ FG 4. 6. 4

89 Multi-Step A rectangular classroom is 10 m long and 4.6 m wide. Make a scale drawing of the classroom using the following scales. 7. 1 cm : 1 m 8. 1 cm : 2 m 9. 1 cm : 2. Given: rectangle MNPQ ∼ rectangle RSTU p. 490 10. Find the perimeter of rectangle RSTU. 11. Find the area of rectangle RSTU. Independent Practice For See Exercises Example 12 13–14 15–17 18–19 1 2 3 4 TEKS TEKS TAKS TAKS Skills Practice p. S17 Application Practice p. S34 PRACTICE AND PROBLEM SOLVING 12. Measurement Jenny is 5 ft 2 in. tall. To find the height of a light pole, she measured her shadow and the pole’s shadow. What is the height of the pole? Space Exploration Use the following information for Exercises 13 and 14. This is a map of the Mars Exploration Rover Opportunity’s predicted landing site on Mars. The scale is 1 cm : 9.4 km. What are the approximate measures of the actual length and width of the ellipse? 13. ̶̶ KJ 14. ̶̶ NP � � � � Multi-Step A park at the end of a city block is a right triangle with legs 150 ft and 200 ft long. Make a scale drawing of the park using the following scales. 15. 1.5 in. : 100 ft 16. 1 in. : 300 ft 17. 1 in. : 150 ft 7- 5 Using Proportional Relationships 491 491 �� Given that pentagon ABCDE ∼ pentagon FGHJK, find each of the following. 18. perimeter of pentagon FGHJK 19. area of pentagon FGHJK Estimation Use the scale on the map for Exercises 20–23. Give the approximate distance of the shortest route between each pair of sites. 20. campfire and the lake 21. lookout point and the campfire 22. cabins and the dining hall 23. lookout point and the lake Given: △ABC ∼ △DEF 24. The ratio of the perimeter of △ABC to the perimeter of △DEF is 8 __ 9. What is the similarity ratio of △ABC to △DEF? 25. The ratio of the area of △

ABC to the area of △DEF is 16 __ 25. What is the similarity ratio of △ABC to △DEF? 26. The ratio of the area of △ABC to the area of △DEF is 4 __ 81. What is the ratio of the perimeter of △ABC to the perimeter of △DEF? 27. Space Exploration The scale of this model of the space shuttle is 1 ft : 50 ft. In the actual space shuttle, the main cargo bay measures 15 ft wide by 60 ft long. What are the dimensions of the cargo bay in the model? 28. Given that △PQR ∼ △WXY, find each ratio. a. perimeter of △PQR__ perimeter of △WXY b. area of △PQR __ area of △WXY c. How does the result in part a compare with the result in part b? 29. Given that rectangle ABCD ∼ EFGH. The area of rectangle ABCD is 135 in 2. The area of rectangle EFGH is 240 in 2. If the width of rectangle ABCD is 9 in., what is the length and width of rectangle EFGH? 30. Sports An NBA basketball court is 94 ft long and 50 ft wide. Make a scale drawing of a court using a scale of 1 __ 4 in. : 10 ft. 31. This problem will prepare you for the Multi-Step TAKS Prep on page 502. A blueprint for a skateboard ramp has a scale of 1 in. : 2 ft. On the blueprint, the rectangular piece of wood that forms the ramp measures 2 in. by 3 in. a. What is the similarity ratio of the blueprint to the actual ramp? b. What is the ratio of the area of the ramp on the blueprint to its actual area? c. Find the area of the actual ramp. 492 492 Chapter 7 Similarity �� Math History In 1075 C.E., Shen Kua created a calendar for the emperor by measuring the positions of the moon and planets. He plotted exact coordinates three times a night for five years. Source: history.mcs. st-andrews.ac.uk 32. Estimation The photo shows a person who is 5

ft 1 in. tall standing by a statue in Jamestown, North Dakota. Estimate the actual height of the statue by using a ruler to measure her height and the height of the statue in the photo. 33. Math History In A.D. 1076, the mathematician Shen Kua was asked by the emperor of China to produce maps of all Chinese territories. Shen created 23 maps, each drawn with a scale of 1 cm : 900,000 cm. How many centimeters long would a 1 km road be on such a map? 34. Points X, Y, and Z are the midpoints of ̶̶ JK, ̶̶ KL, and ̶̶ LJ, respectively. What is the ratio of the area of △JKL to the area of △XYZ? 35. Critical Thinking Keisha is making two scale drawings of her school. In one drawing, she uses a scale of 1 cm : 1 m. In the other drawing, she uses a scale of 1 cm : 5 m. Which of these scales will produce a smaller drawing? Explain. 36. The ratio of the perimeter of square ABCD to the perimeter of square EFGH is 4 __ 9. Find the side lengths of each square. 37. Write About It Explain what it would mean to make a scale drawing with a scale of 1 : 1. 38. Write About It One square has twice the area of another square. Explain why it is impossible for both squares to have side lengths that are whole numbers. 39. △ABC ∼ △RST, and the area of △ABC is 24 m 2. What is the area of △RST? 16 m 2 29 m 2 36 m 2 54 m 2 40. A blueprint for a museum uses a scale of 1 __ in. : 1 ft. 4 One of the rooms on the blueprint is 3 3 __ in. long. 4 How long is the actual room? 4 ft 15 ft 45 ft 180 ft 41. The similarity ratio of two similar pentagons is 9 __. What is the ratio of the 4 perimeters of the pentagons 81 _ 16 42. Of two similar triangles, the second triangle has sides half the length of the first. Given that the area of the first triangle is 16 ft 2, find the area of the second. 32 ft 2 16 ft 2 4 ft 2 8 ft 2 7- 5 Using Proportional Relationships 493 493 �� CHALLENGE

AND EXTEND 43. Astronomy The city of Eugene, Oregon, has a scale model of the solar system nearly 6 km long. The model’s scale is 1 km : 1 billion km. a. Earth is 150,000,000 km from the Sun. How many meters apart are Earth and the Sun in the model? b. The diameter of Earth is 12,800 km. What is the diameter, in centimeters, of Earth in the model? �� 44. Given: △ABC ∼ △DEF AB + BC + AC __ DE + EF +DF Prove: = AB _ DE 45. Given: △PQR ∼ △WXY Area △PQR __ Area △WXY Prove: = PR 2 _ WY 2 46. Quadrilateral PQRS has side lengths of 6 m, 7 m, 10 m, and 12 m. The similarity ratio of quadrilateral PQRS to quadrilateral WXYZ is 1 : 2. a. Find the lengths of the sides of quadrilateral WXYZ. b. Make a table of the lengths of the sides of both figures. c. Graph the data in the table. d. Determine an equation that relates the lengths of the sides of quadrilateral PQRS to the lengths of the sides of quadrilateral WXYZ. SPIRAL REVIEW Solve each equation. Round to the nearest hundredth if necessary. (Previous course) 47. (x - 3) 2 = 49 49. 4 (x + 2) 2 - 28 = 0 48. (x + 1) 2 - 4 = 0 Show that the quadrilateral with the given vertices is a parallelogram. (Lesson 6-3) 50. A (-2, -2), B (1, 0), C (5, 0), D (2, -2) 51. J (1, 3), K (3, 5), L (6, 2), M (4, 0) 52. Given that 58x = 26y, find the ratio y : x in simplest form. (Lesson 7-1) KEYWORD: MG7 Career Q: What math classes did you take in high school? A: Algebra, Geometry, and Probability and Statistics Q: What math-related classes did you take in college? A: Trigonometry, Precalculus, Drafting

, and System Design Q: How do photogrammetrists use math? A: Photogrammetrists use aerial photographs to make detailed maps. To prepare maps, I use computers and perform a lot of scale measures to make sure the maps are accurate. Q: What are your future plans? A: My favorite part of making maps is designing scale drawings. Someday I’d like to apply these skills toward architectural work. Elaine Koch Photogrammetrist 494 494 Chapter 7 Similarity �� 7-6 Dilations and Similarity in the Coordinate Plane TEKS G.11.A Similarity and the geometry of shape: use... properties and transformations to... justify conjectures.... Objectives Apply similarity properties in the coordinate plane. Use coordinate proof to prove figures similar. Vocabulary dilation scale factor Also G.2.B, G.9.B Who uses this? Computer programmers use coordinates to enlarge or reduce images. Many photographs on the Web are in JPEG format, which is short for Joint Photographic Experts Group. When you drag a corner of a JPEG image in order to enlarge it or reduce it, the underlying program uses coordinates and similarity to change the image’s size. A dilation is a transformation that changes the size of a figure but not its shape. The preimage and the image are always similar. A scale factor describes how much the figure is enlarged or reduced. For a dilation with scale factor k, you can find the image of a point by multiplying each coordinate by k: (a, b) → (ka, kb). E X A M P L E 1 Computer Graphics Application If the scale factor of a dilation is greater than 1 (k > 1), it is an enlargement. If the scale factor is less than 1 (k < 1), it is a reduction. The figure shows the position of a JPEG photo. Draw the border of the photo after a dilation with scale factor 3 __. 2 Step 1 Multiply the vertices of the photo A (0, 0), B (0, 4), C (3, 4), and D (3, 0) by 3 __ 2. Rectangle ABCD Rectangle A'B'C'D' _ _ A (0, 0) → A' (0 ⋅ 3 ) → A' (0, 00, 4) → B' (0 ⋅ 3 ) → B' (0, 63, 4)

→ C' (3 ⋅ 3 ) → C' (4.5, 63, 0) → D' (3 ⋅ 3 ) → D' (4.5, 0), 0 ⋅ 3 2 2 Step 2 Plot points A' (0, 0), B' (0, 6), C' (4.5, 6), and D' (4.5, 0). Draw the rectangle. 1. What if…? Draw the border of the original photo after a dilation with scale factor 1 __ 2. 7- 6 Dilations and Similarity in the Coordinate Plane 495 495 �� E X A M P L E 2 Finding Coordinates of Similar Triangles Given that △AOB ∼ △COD, find the coordinates of D and the scale factor. Since △AOB ∼ △COD, AO _ = OB _ CO OD = 3 _ 2 _ 4 OD Substitute 2 for AO, 4 for CO, and 3 for OB. 2OD = 12 OD = 6 Cross Products Prop. Divide both sides by 2. D lies on the x-axis, so its y-coordinate is 0. Since OD = 6, its x-coordinate must be 6. The coordinates of D are (6, 0). (3, 0) → (3 ⋅ 2, 0 ⋅ 2) → (6, 0), so the scale factor is 2. 2. Given that △MON ∼ △POQ and coordinates P (-15, 0), M (-10, 0), and Q (0, -30), find the coordinates of N and the scale factor. E X A M P L E 3 Proving Triangles Are Similar Given: A (1, 5), B (-1, 3), C (3, 4), D (-3, 1), and E (5, 3) Prove: △ABC ∼ △ADE Step 1 Plot the points and draw the triangles. Step 2 Use the Distance Formula to find the side lengths. AB = √  (-1 - 1) 2 + (3 - 5) 2 AC = √  (3 - 1

) 2 + (4 - 5 AD = √  (-3 - 1) 2 + (1 - 5) 2 AE = √  (5 - 1) 2 + (3 - 5) 2 = √  32 = 4 √  2 = √  20 = 2 √  5 Step 3 Find the similarity ratio. AB _ AD = AC _ AE √  Since AB ___ AD by SAS ∼. = AC ___ AE and ∠A ≅ ∠A by the Reflexive Property, △ABC ∼ △ADE 3. Given: R (-2, 0), S (-3, 1), T (0, 1), U (-5, 3), and V (4, 3) Prove: △RST ∼ △RUV 496 496 Chapter 7 Similarity �� E X A M P L E 4 Using the SSS Similarity Theorem Graph the image of △ABC after a dilation with scale factor 2. Verify that △A'B'C'∼ △ABC. Step 1 Multiply each coordinate by 2 to find the coordinates of the vertices of △A'B'C '. A (2, 3) → A' (2 ⋅ 2, 3 ⋅ 2) = A' (4, 6) B (0, 1) → B' (0 ⋅ 2, 1 ⋅ 2) = B' (0, 2) C (3, 0) → C' (3 ⋅ 2, 0 ⋅ 2) = C' (6, 0) Step 2 Graph △A'B'C '. Step 3 Use the Distance Formula to find the side lengths. AB = √  (2 - 0) 2 + (3 - 1) 2 A'B' = √  (4 - 0) 2 + (6 - 2 =

√  32 = 4 √  2 BC = √  (3 - 0) 2 + (0 - 1) 2 B'C'= √  (6 - 0) 2 + (0 - 2) 2 = √  10 = √  40 = 2 √  10 AC = √  (3 - 2) 2 + (0 - 3) 2 A'C'= √  (6 - 4) 2 + (0 - 6) 2 = √  10 = √  40 = 2 √  10 Step 4 Find the similarity ratio. A'B' _ AB =, B'C'_ = BC 2 √  10 _ √  10 = 2, A'C'_ = AC 2 √  10 _ √  10 = 2 Since A'B' _ = B'C' _ BC = A'C' _ AC AB, △ABC ∼ △A'B'C'by SSS ∼. 4. Graph the image of △MNP after a dilation with scale factor 3. Verify that △M'N'P' ∼ △MNP. THINK AND DISCUSS 1. △JKL has coordinates J (0, 0), K (0, 2), and L (3, 0). Its image after a dilation has coordinates J' (0, 0), K'(0, 8), and L' (12, 0). Explain how to find the scale factor of the dilation. 2. GET ORGANIZED Copy and complete the graphic organizer. Write the definition of a dilation, a property of dilations, and an example and nonexample of a dilation. 7- 6 Dilations and Similarity in the Coordinate Plane 497 497 ��

�� 7-6 Exercises Exercises KEYWORD: MG7 7-6 KEYWORD: MG7 Parent GUIDED PRACTICE Vocabulary Apply the vocabulary from this lesson to answer each question. 1. A? is a transformation that proportionally reduces or enlarges a figure, ̶̶̶̶ such as the pupil of an eye. (dilation or scale factor) 2. A ratio that describes or determines the dimensional relationship of a figure to that which it represents, such as a map scale of 1 in. : 45 ft, is called a (dilation or scale factor)?. ̶̶̶̶. Graphic Design A designer created p. 495 this logo for a real estate agent but needs to make the logo twice as large for use on a sign. Draw the logo after a dilation with scale factor 2. Given that △AOB ∼ △COD, p. 496 find the coordinates of C and the scale factor. 5. Given that △ROS ∼ △POQ, find the coordinates of S and the scale factor. Given: A (0, 0), B (-1, 1), C (3, 2), D (-2, 2), and E (6, 4) p. 496 Prove: △ ABC ∼ △ADE 7. Given: J (-1, 0), K (-3, -4), L (3, -2), M (-4, -6), and N (5, -3) Prove: △ JKL ∼ △JMN. 497 Multi-Step Graph the image of each triangle after a dilation with the given scale factor. Then verify that the image is similar to the given triangle. 9. scale factor 3 __ 2 8. scale factor 2 498 498 Chapter 7 Similarity xy440ge07se_c07l06004a�� Independent Practice For See Exercises Example 10 11–12 13–14 15–16 1 2 3 4 TEKS TEKS TAKS TAKS Skills Practice p. S17 Application Practice p. S34 PRACTICE AND PROBLEM SOLVING 10. Advertising A promoter produced this design for a street festival. She now wants to make the design smaller

to use on postcards. Sketch the design after a dilation with scale factor 1 __ 2. 11. Given that △UOV ∼ △XOY, find the coordinates of X and the scale factor. 12. Given that △MON ∼ △KOL, find the coordinates of K and the scale factor. 13. Given: D (-1, 3), E (-3, -1), F (3, -1), G (-4, -3), and H (5, -3) Prove: △DEF ∼ △DGH 14. Given: M (0, 10), N (5, 0), P (15, 15), Q (10, -10), and R (30, 20) Prove: △MNP ∼ △MQR Multi-Step Graph the image of each triangle after a dilation with the given scale factor. Then verify that the image is similar to the given triangle. 15. J (-2, 0) and K (-1, -1), and L (-3, -2) with scale factor 3 16. M (0, 4), N (4, 2), and P (2, -2) with scale factor 1 __ 2 17. Critical Thinking Consider the transformation given by the mapping (x, y) → (2x, 4y). Is this transformation a dilation? Why or why not? 18. /////ERROR ANALYSIS///// Which solution to find the scale factor of the dilation that maps △RST to △UVW is incorrect? Explain the error. 19. Write About It A dilation maps △ABC to △A'B 'C '. How is the scale factor of the dilation related to the similarity ratio of △ABC to △A'B 'C '? Explain. 20. This problem will prepare you for the Multi-Step TAKS Prep on page 502. a. In order to build a skateboard ramp, Miles draws △JKL on a coordinate plane. One unit on the drawing represents 60 cm of actual distance. Explain how he should assign coordinates for the vertices of △JKL. b. Graph the image of △JKL after a dilation with scale factor 3. 7- 6 Dilations and Similarity in the Coordinate Plane 499 499 yxc07106005a48840��

�� 21. Which coordinates for C make △COD similar to △AOB? (0, 2.4) (0, 2.5) (0, 3) (0, 3.6) 22. A dilation with scale factor 2 maps △RST to △R'S'T'. The perimeter of △RST is 60. What is the perimeter of △R'S'T'? 30 60 120 240 23. Which triangle with vertices D, E, and F is similar to △ABC? D (1, 2), E (3, 2), F (2, 0) D (-1, -2), E (2, -2), F (1, -5) D (1, 2), E (5, 2), F (3, 0) D (-2, -2), E (0, 2), F (-1, 0) 24. Gridded Resonse ̶̶ AB with endpoints A (3, 2) and B (7, 5) is dilated by a scale factor of 3. Find the length of ̶̶̶ A'B'. CHALLENGE AND EXTEND 25. How many different triangles having ̶̶ XY as a side are similar to △MNP? 26. △XYZ ∼ △MPN. Find the coordinates of Z. 27. A rectangle has two of its sides on the x- and y-axes, a vertex at the origin, and a vertex on the line y = 2x. Prove that any two such rectangles are similar. 28. △ ABC has vertices A (0, 1), B (3, 1), and C (1, 3). △DEF has vertices D (1, -1) and E (7, -1). Find two different locations for vertex F so that △ABC ∼ △DEF. SPIRAL REVIEW Write an inequality to represent the situation. (Previous

course) 29. A weight lifter must lift at least 250 pounds. There are two 50-pound weights on a bar that weighs 5 pounds. Let w represent the additional weight that must be added to the bar. Find the length of each segment, given that (Lesson 5-2) ̶̶ HF 30. 31. ̶̶ JF ̶̶ DE ≅ ̶̶ FE. 32. ̶̶ CF △SUV ∼ △SRT. Find the length of each segment. (Lesson 7-4) 33. ̶̶ RT 34. ̶̶ V T 35. ̶̶ ST 500 500 Chapter 7 Similarity �� Direct Variation Algebra In Lesson 7-6 you learned that for two similar figures, the measure of each point was multiplied by the same scale factor. Is the relationship between the scale factor and the perimeter of the figure a direct variation? y Recall from algebra that if y varies directly as x, then y = kx, or where k is the constant of variation. __ x = k, See Skills Bank page S62 Example A rectangle has a length of 4 ft and a width of 2 ft. Find the relationship between the scale factors of similar rectangles and their corresponding perimeters. If the relationship is a direct variation, find the constant of variation. Step 1 Make a table to record data. Scale Factor k 1 _ 2 2 3 4 5 Length ℓ = k (4) ℓ = 1 _ (4) = 2 2 Width w = k (2) w = 1 _ (2) = 1 2 Perimeter P = 2ℓ + 2w 2 (2) + 2 (1) = 6 8 12 16 20 4 6 8 10 24 36 48 60 Step 2 Graph the points ( 1 _, 6), (2, 24), (3, 36), (4, 48), and (5, 60). 2 Since the points are collinear and the line that contains them includes the origin, the relationship is a direct variation. Step 3 Find the equation of direct variation. y = kx 60 = k (5) Substitute 60 for y and 5 for x. 12 = k Divide both sides by 5. y = 12 x Substitute 12 for k. Thus the constant of variation is 12. Try This TAKS Grades 9–

11 Obj. 3, 8, 10 Use the scale factors given in the above table. Find the relationship between the scale factors of similar figures and their corresponding perimeters. If the relationship is a direct variation, find the constant of variation. 1. regular hexagon with side length 6 2. triangle with side lengths 3, 6, and 7 3. square with side length 3 On Track for TAKS 501 501 �� SECTION 7B Applying Similarity Ramp It Up Many companies sell plans for build-it-yourself skateboard ramps. The figures below show a ramp and the plan for the triangular support structure at the ̶̶̶ GH, side of the ramp. In the plan, ̶̶ BC. and ̶̶ EF, ̶̶ JK are perpendicular to the base ̶̶ AB, 1. The instructions call for extra pieces of wood to ̶̶ GJ, and ̶̶ JC. Given AE = 42.2 cm, reinforce find EG, GJ, and JC to the nearest tenth. ̶̶ EG, ̶̶ AE, 2. Once the support structure is built, it is covered with a triangular piece of plywood. Find the area of the piece of wood needed to cover △ABC. A separate blueprint for the ramp uses a scale of 1 cm : 25 cm. What is the area of △ABC in the blueprint? 3. Before building the ramp, you transfer the plan to a coordinate plane. Draw △ABC on a coordinate plane so that 1 unit represents 25 cm and B is at the origin. Then draw the image of △ABC after a dilation with scale factor 3 __ 2. 502 502 Chapter 7 Similarity �� Quiz for Lessons 7-4 Through 7-6 SECTION 7B 7-4 Applying Properties of Similar Triangles Find the length of each segment. ̶̶ ST 1. ̶̶ AB and ̶̶ AC 2. 3. An artist drew a picture of railroad tracks ̶̶ EF, such that the ties What is the length of ̶̶̶ GH, and ̶̶ FH? ̶̶ JK are parallel. 7-5 Using Proportional Relationships The plan for a restaurant uses the scale of 1.5 in. : 60 ft. Find the actual length of the following walls. ̶̶ AB ̶̶ CD 4

. 6. ̶̶ BC ̶̶ EF 5. 7. 8. A student who is 5 ft 3 in. tall measured her shadow and the shadow cast by a water tower shaped like a golf ball. What is the height of the tower? 7-6 Dilations and Similarity in the Coordinate Plane 9. Given: A (-1, 2), B (-3, -2), C (3, 0), D (-2, 0), and E (1, 1) Prove: △ADE ∼ △ABC 10. Given: R (0, 0), S (-2, -1), T (0, -3), U (4, 2), and V (0, 6) Prove: △RST ∼ △RUV Graph the image of each triangle after a dilation with the given scale factor. Then verify that the image is similar to the given triangle. 11. scale factor 3 12. scale factor 1.5 Ready to Go On? 503 503 ��ge07sec07rg2001aaAB40 ft5 ft 10 in.�� For a complete list of the postulates and theorems in this chapter, see p. S82. Vocabulary cross products.............. 455 proportion................. 455 scale factor................. 495 dilation.................... 495 ratio....................... 454 similar..................... 462 extremes................... 455 scale....................... 489 similar polygons............ 462 indirect measurement....... 488 scale drawing............... 489 similarity ratio.........

.... 463 means..................... 455 Complete the sentences below with vocabulary words from the list above. 1. An equation stating that two ratios are equal is called a(n)?. ̶̶̶̶? is a transformation that changes the size of a figure but not its shape. ̶̶̶̶ 2. A(n) 3. In the proportion u _ v = x _ y, the 4. A(n)? compares two numbers by division. ̶̶̶̶? are v and x. ̶̶̶̶ 7-1 Ratio and Proportion (pp. 454–459) TEKS G.5.B, G.7.B, G.7.C, G.11.B E X A M P L E S EXERCISES ■ Write a ratio expressing the slope of ℓ. slope = rise _ run 1 - 3 = = 2 _ -4 = - 1 _ 2 Write a ratio expressing the slope of each line. 5. m 6. n 7. p ■ Solve the proportion. _ x - 3 50 2 _ = 4 (x - 3) 2 = 2 (50) 4 (x - 3) 2 4 (x - 3) = 100 2 (x - 3) = 25 x - 3 = ±5 Cross Products Prop. Simplify. Divide both sides by 4. Find the square root of both sides. x - 3 = 5 or x - 3 = -5 Rewrite as two eqns. x = 8 or x = -2 Add 3 to both sides. 504 504 Chapter 7 Similarity 8. If 84 is divided into three parts in the ratio 3 : 5 : 6, what is the sum of the smallest and the largest part? 9. The ratio of the measures of a pair of sides of a rectangle is 7 : 12. If the perimeter of the rectangle is 95, what is the length of each side? Solve each proportion. y = 9 _ _ 10. 7 3 = 9 _ x 12. x _ 4 = 3x _ 14. 12 _ 32 2x = 25 _ s 11. 10 _ 4 13 24 15. = = z - 1 _ 36 2 _ 3 (y + 1) �� 7-2 Ratios in Similar Polygons (pp. 462–467) TEKS G.5

.B, G.11.A, G.11.B E X A M P L E EXERCISES ■ Determine whether △ABC and △DEF are similar. If so, write the similarity ratio and a similarity statement. Determine whether the polygons are similar. If so, write the similarity ratio and a similarity statement. 16. rectangles JKLM and PQRS 17. △TUV and △WXY It is given that ∠A ≅ ∠D and ∠B ≅ ∠E. ∠C ≅ ∠F by the Third Angles Theorem. AB ___ = AC ___ DE DF is 2 __ 3, and △ABC ∼ △DEF. = 2 __ 3. Thus the similarity ratio = BC ___ EF 7-3 Triangle Similarity: AA, SSS, and SAS (pp. 470–477) TEKS G.5.B, G.9.B, G.11.A, G.11.B E X A M P L E ■ Given: ̶̶ AB ǁ ̶̶ CD, AB = 2CD, AC = 2CE Prove: △ABC ∼ △CDE Proof: Statements Reasons ̶̶ AB ǁ ̶̶ CD 1. 1. Given 2. ∠BAC ≅ ∠DCE 2. Corr.  Post. 3. AB = 2CD, AC = 2CE 3. Given = 2 4. AB ___ CD 5. AB ___ CD = 2, AC ___ CE = AC ___ CE 6. △ABC ∼ △CDE 4. Division Prop. 5. Trans. Prop. of = 6. SAS ∼ (Steps 2, 5) EXERCISES 18. Given: JL = 1 _ JN, JK = 1 _ 3 3 Prove: △JKL ∼ △JMN JM 19. Given: ̶̶ QR ǁ ̶̶ ST Prove: △PQR ∼ △PTS 20. Given: ̶̶ BD ǁ ̶̶ CE Prove: AB (CE) = AC (BD) (Hint: After you have proved the triangles similar, look for a proportion using AB, AC, CE, and BD, the lengths of corresponding sides.) Study Guide: Review 505 505 ��

�� 7-4 Applying Properties of Similar Triangles (pp. 481–487) TEKS G.2.A, G.3.B, G.5.B, G.9.B, G.11.A, G.11.B EXERCISES Find each length. 21. CE 22. ST Verify that the given segments are parallel. 23. ̶̶ KL and ̶̶̶ MN 24. ̶̶ AB and ̶̶ CD 25. Find SU and SV. 26. Find the length of the third side of △ABC. 27. One side of a triangle is x inches longer than another side. The ray bisecting the angle formed by these sides divides the opposite side into 3-inch and 5-inch segments. Find the perimeter of the triangle in terms of x. E X A M P L E S ■ Find PQ. ̶̶ = PR ___ QR ǁ It is given that RT Triangle Proportionality Theorem. ̶̶ ST, so PQ ___ QS by the PQ = 15 _ _ 5 6 Substitute 5 for QS, 15 for PR, and 6 for RT. 6 (PQ) = 75 Cross Products Prop. PQ = 12.5 Divide both sides by 6. ̶̶ AB ǁ ̶̶ CD. = 1.5 EC _ CA ED _ DB ■ Verify that = 6 _ 4 = 4.5 _ 3 Since EC ___ = ED ___, DB CA = 1.5 ̶̶ AB ǁ ̶̶ CD by the Converse of the Triangle Proportionality Theorem. ■ Find JL and LK. Since ̶̶ JK bisects ∠LJM, JL ___ LK = JM ___ MK by the Triangle Angle Bisector Theorem. 3x - 2 _ 2x = 12.5 _ 10 Substitute the given values. 10 (3x - 2) = 12.5 (2x) Cross Products Prop. 30x - 20 = 25x Simplify. 30x = 25x + 20 Add 20 to both sides. 5x = 20 Subtract 25x from both sides. x = 4 Divide both sides by 5. JL = 3x - 2 = 3 (4) - 2 = 10 LK = 2x = 2 (4) = 8 506 506 Chapter 7 Similarity ��

�� 7-5 Using Proportional Relationships (pp. 488–494) TEKS G.1.B, G.5.A, G.11.A, G.11.B, G.11.D E X A M P L E EXERCISES ■ Use the dimensions in the diagram to find the height h of the tower. A student who is 5 ft 5 in. tall measured his shadow and a tower’s shadow to find the height of the tower. 28. To find the height of a flagpole, Casey measured her own shadow and the flagpole’s shadow. Given that Casey’s height is 5 ft 4 in., what is the height x of the flagpole? 5 ft 5 in. = 65 in. 1 ft 3 in. = 15 in. 11 ft 3 in. = 135 in. h _ 135 = 65 _ 15 15h = 65 (135) 15h = 8775 h = 585 in. Corr. sides are proportional. Cross Products Prop. Simplify. Divide both sides by 15. The height of the tower is 48 ft 9 in. 29. Jonathan is 3 ft from a lamppost that is 12 ft high. The lamppost and its shadow form the legs of a right triangle. Jonathan is 6 ft tall and is standing parallel to the lamppost. How long is Jonathan’s shadow? 7-6 Dilations and Similarity in the Coordinate Plane (pp. 495–500) E X A M P L E EXERCISES TEKS G.2.B, G.9.B, G.11.A ■ Given: A (5, -4), B (-1, -2), C (3, 0), D (-4, -1) and E (2, 2) Prove: △ABC ∼ △ADE Proof: Plot the points and draw the triangles. 30. Given: R (1, -3), S (-1, -1), T (2, 0), U (-3, 1), and V (3, 3) Prove: △RST ∼ △RUV 31. Given: J (4, 4), K (2, 3), L (4, 2), M (-4, 0), and N (4, -

4) Prove: △JKL ∼ △JMN 32. Given that △AOB ∼ △COD, find the coordinates of B and the scale factor. Use the Distance Formula to find the side lengths. AC = 2 √  5, AE = 3 √  5 AB = 2 √  10, AD = 3 √  10 = 2 _. 3 Therefore AB _ AD = AC _ AE Since corresponding sides are proportional and ∠A ≅ ∠A by the Reflexive Property, △ABC ∼ △ADE by SAS ∼. 33. Graph the image of the triangle after a dilation with the given scale factor. Then verify that the image is similar to the given triangle. K (0, 3), L (0, 0), and M (4, 0) with scale factor 3. Study Guide: Review 507 507 �� 1. Two points on ℓ are A (-6, 4) and B (10, -6). Write a ratio expressing the slope of ℓ. 2. Alana has a photograph that is 5 in. long and 3.5 in. wide. She enlarges it so that its length is 8 in. What is the width of the enlarged photograph? Determine whether the polygons are similar. If so, write the similarity ratio and a similarity statement. 3. △ABC and △MNP 4. rectangle DEFG and rectangle HJKL 5. Given: RSTU Prove: △RWV ∼ △SWT 6. Derrick is building a skateboard ramp as shown. Given that BD = DF = FG = 3 ft, find CD and EF to the nearest tenth. Find the length of each segment. 7. ̶̶ PR ̶̶̶ YW and ̶̶̶ WZ 8. 9. To find the height of a tree, a student measured the tree’s shadow and her own shadow. If the student’s height is 5 ft 8 in., what is the height of the tree? 10. The plan for a living room uses the scale of 1.5 in. : 30 ft. Use a ruler and find the length of the actual room’s diagonal ̶̶ AB. ��

�� 11. Given: A (6, 5), B (3, 4), C (6, 3), D (-3, 2), and E (6, -1) Prove: △ABC ∼ △ADE 12. A quilter designed this patch for a quilt but needs a larger version for a different project. Draw the quilt patch after a dilation with scale factor 3 __ 2. 508 508 Chapter 7 Similarity �� FOCUS ON SAT The SAT consists of seven test sections: three verbal, three math, and one more verbal or math section not used to compute your final score. The “extra” section is used to try out questions for future tests and to compare your score to previous tests. Read each question carefully and make sure you answer the question being asked. Check that your answer makes sense in the context of the problem. If you have time, check your work. You may want to time yourself as you take this practice test. It should take you about 8 minutes to complete. 1. In the figure below, the coordinates of the vertices are A (1, 5), B (1, 1), D (10, 1), and E (10, -7). If the length of the coordinates of C? ̶̶ CE is 10, what are Note: Figure not drawn to scale. (A) (4, 1) (B) (1, 4) (C) (7, 1) (D) (1, 7) (E) (6, 1) 2. In the figure below, triangles JKL and MKN are similar, and ℓ is parallel to segment JL. What is the length of ̶̶̶ KM? Note: Figure not drawn to scale. (A) 4 (B) 8 (C) 9 (D) 13 (E) 18 3. Three siblings are to share an inheritance of $750,000 in the ratio 4 : 5 : 6. What is the amount of the greatest share? (A) $125,000 (B) $187,500 (C) $250,000 (D) $300,000 (E) $450,000 4. A 35-foot flagpole casts a 9-foot shadow at the same time that a girl casts a 1

.2-foot shadow. How tall is the girl? (A) 3 feet 8 inches (B) 4 feet 6 inches (C) 4 feet 7 inches (D) 4 feet 8 inches (E) 5 feet 6 inches 5. What polygon is similar to every other polygon of the same name? (A) Triangle (B) Parallelogram (C) Rectangle (D) Square (E) Trapezoid College Entrance Exam Practice 509 509 �� Any Question Type: Interpret A Diagram When a diagram is included as part of a test question, do not make any assumptions about the diagram. Diagrams are not always drawn to scale and can be misleading if you are not careful. Multiple Choice What is DE? 3.6 4 4.8 9 Make your own sketch of the diagram. Separate the two triangles so that you are able to find the side length measures. By redrawing the diagram, it is clear that the two triangles are similar. Set up a proportion to find DE. AB _ BC 6 _ 10 48 _ 10 DE = 4.8 = DE _ EF = DE _ 8 = DE The correct choice is C. Gridded Response △X′ Y ′Z′ is the image of △XYZ after a dilation with scale factor 1 __. Find X ′Z′. 2 Before you begin, look at the scale of both the x-axis and the y-axis. Do not assume that the scale is always 1. At first glance, you might assume that XZ is 4. But by looking closely at the x-axis, notice that each increment represents 2 units. So XZ is actually 8. When △XYZ is dilated by a factor of 1 _ 2, X′Z′ will be half of XZ. XZ = 1 _ X′Z′ = 1 _ 2 2 (8) = 4 510 510 Chapter 7 Similarity �� If the diagram does not match the given information, draw one that is more accurate. Item C Gridded Response What is the measure of MN? Read each test item and answer the questions that follow. Item A Multiple Choice Which ratio is the slope of m? 1 _ 15 1 _ 3 3 15 1. What is the scale of the y-axis? Use this scale to determine the rise of the slope. 2. What is the scale of the x-axis? Use

this scale to determine the run of the slope. 3. Write the ratio that represents the slope of m. 4. Anna selected choice B as her answer. Is she correct? If not, what do you think she did wrong? Item B Gridded Response If ABCD ∼ MNOP and AC is 6, what is AB? 5. Examine the figures. Do you think longer or shorter than ̶̶̶ MN? ̶̶ AB is 8. Describe how redrawing the figure can help you better understand the given information. 9. After reading this test question, a student redrew the figure as shown below. Explain if it is a correct interpretation of the original figure. If it is not, redraw and/or relabel it so that it is correct. Item D Multiple Choice Which is a similarity ratio for the triangles shown? 20 _ 1 10 _ 1 2 _ 1 15 _ 1 6. Do you think the drawings actually represent the given information? If not, explain why. 7. Create your own sketch of the figures to more accurately match the given information. 10. Chad determined that choice D was correct. Do you agree? If not, what do you think he did wrong? 11. Redraw the figures so that they are easier to understand. Write three statements that describe which vertices correspond to each other and three statements that describe which sides correspond to each other. TAKS Tackler 511 511 �� KEYWORD: MG7 TestPrep CUMULATIVE ASSESSMENT, CHAPTERS 1–7 Multiple Choice 1. Which similarity statement is true for rectangles ABCD and MNPQ, given that AB = 3, AD = 4, MN = 6, and NP = 4.5? Rectangle ABCD ∼ rectangle MNPQ 5. If 12x = 16y, what is the ratio of x to y in simplest form Rectangle ABCD ∼ rectangle PQMN Use the diagram for Items 6 and 7. Rectangle ABCD ∼ rectangle MPNQ Rectangle ABCD ∼ rectangle QMNP 2. △ ABC has perpendicular bisectors If AP = 6 and ZP = 4.5, what is the length of to the nearest tenth? ̶̶ XP, ̶̶ YP, and ̶̶ BC ̶̶ ZP. 6. Given that ̶̶ AB ≅ ̶̶ CD, which additional information would

be sufficient to prove that ABCD is a parallelogram? ̶̶ AB ǁ ̶̶ AC ǁ ̶̶ CD ̶̶ BD ∠CAB ≅ ∠CDB E is the midpoint of ̶̶̶ AD. 7. If   AC is parallel to   BD and m∠1 + m∠2 = 140°, what is the measure of ∠3? 20° 40° 50° 70° 8. If of ̶̶ AC is twice as long as ̶̶ DC? ̶̶ AB, what is the length 2.5 centimeters 3.75 centimeters 5 centimeters 15 centimeters 4.0 7.9 9.0 12.7 3. What is the converse of the statement “If a quadrilateral has 4 congruent sides, then it is a rhombus”? If a quadrilateral is a rhombus, then it has 4 congruent sides. If a quadrilateral does not have 4 congruent sides, then it is not a rhombus. If a quadrilateral is not a rhombus, then it does not have 4 congruent sides. If a rhombus has 4 congruent sides, then it is a quadrilateral. 4. A blueprint for a hotel uses a scale of 3 in. : 100 ft. On the blueprint, the lobby has a width of 1.5 in. and a length of 2.25 in. If the carpeting for the lobby costs $1.25 per square foot, how much will the carpeting for the entire lobby cost? $312.50 $1406.25 $3000.00 $4687.50 512 512 Chapter 7 Similarity �� When writing proportions for similar figures, make sure that each ratio compares corresponding side lengths in each figure. STANDARDIZED TEST PREP Short Response 17. △ ABC has vertices A (-2, 0), B (2, 2), and C (2, -2). △DEC has vertices D (0, -1), E (2, 0), and C (2, -2). Prove that △ ABC ∼ △DEC. 9. What type of triangle has angles that measure (2x) °, (3x - 9)

°, and (x + 27) °? 18. ∠TUV in the diagram below is an obtuse angle. Isosceles acute triangle Isosceles right triangle Scalene acute triangle Scalene obtuse triangle Use the diagram for Items 10 and 11. 10. Which of these points is the orthocenter of △FGH? F G H J Write an inequality showing the range of possible measurements for ∠TUW. Show your work or explain your answer. 19. △ ABC and △ ABD share side ̶̶ AB. Given that △ABC ∼ △ABD, use AAS to explain why these two triangles must also be congruent. 20. Rectangle ABCD has a length of 2.6 cm and a width of 1.8 cm. Rectangle WXYZ has a length of 7.8 cm and a width of 5.4 cm. Determine whether rectangle ABCD is similar to rectangle WXYZ. Explain your reasoning. 11. Which of the following could be the side lengths 21. If △ABC and △XYZ are similar triangles, there are of △FGH? FG = 2, GH = 3, and FH = 4 FG = 4, GH = 5, and FH = 6 FG = 5, GH = 4, and FH = 3 FG = 6, GH = 8, and FH = 10 12. The measure of one of the exterior angles of a right triangle is 120°. What are the measures of the acute interior angles of the triangle? 30° and 60° 40° and 50° 40° and 80° 60° and 60° Gridded Response 13. The ratio of a football field’s length to its width is 9 : 4. If the length of the field is 360 ft, what is the width of the field in feet? 14. The sum of the measures of the interior angles of a convex polygon is 1260°. How many sides does the polygon have? 15. In kite PQRS, ∠P and ∠R are opposite angles. If m∠P = 25° and m∠R = 75°, what is the measure of ∠Q in degrees? 16. Heather is 1.6 m tall and casts a shadow of 3.5 m. At the same time, a barn casts a shadow of 17.5 m. Find the height of the barn

in meters. six possible similarity statements. a. What is the probability that △ABC ∼ △XYZ is correct? b. If △ABC and △XYZ are isosceles, what is the probability that △ABC ∼ △XYZ? c. If △ABC and △XYZ are equilateral, what is the probability that △ABC ∼ △XYZ? Explain. Extended Response 22. a. Given: △SRT ∼ △VUW and ̶̶ VU ≅ ̶̶̶ VW Prove: ̶̶ SR ≅ ̶̶ ST b. Explain in words how you determine the possible values for x and y that would make the two triangles below similar. Note: Triangles not drawn to scale. c. Explain why x cannot have a value of 1 if the two triangles in the diagram above are similar. Cumulative Assessment, Chapters 1–7 513 513 �� Right Triangles and Trigonometry 8A Trigonometric Ratios 8-1 Similarity in Right Triangles Lab Explore Trigonometric Ratios 8-2 Trigonometric Ratios 8-3 Solving Right Triangles 8B Applying Trigonometric Ratios 8-4 Angles of Elevation and Depression Lab Indirect Measurement Using Trigonometry 8-5 Law of Sines and Law of Cosines 8-6 Vectors Ext Trigonometry and the Unit Circle KEYWORD: MG7 ChProj Standing over 567 feet tall, the San Jacinto Monument in LaPorte is the tallest memorial column in the world. 514 514 Chapter 8 Vocabulary Match each term on the left with a definition on the right. 1. altitude A. a comparison of two numbers by division 2. proportion 3. ratio 4. right triangle B. a segment from a vertex to the midpoint of the opposite side of a triangle C. an equation stating that two ratios are equal D. a perpendicular segment from the vertex of a triangle to a line containing the base E. a triangle that contains a right angle Identify Similar Figures Determine if the two triangles are similar. 5. 6. Special Right Triangles Find the value of x. Give the answer in simplest radical form. 7. 8. 9. 10. Solve Multi-Step Equations Solve each equation. 11. 3 (x - 1) = 12 13. 6 = 8 (x

- 3) 12. -2 (y + 5) = -1 14. 2 = -1 (z + 4) Solve Proportions Solve each proportion. 15. 4_ y = 6_ 18 16. 5_ 8 = x_ 32 17. m_ 9 = 8_ 12 18. y_ 4 = 9_ y Rounding and Estimation Round each decimal to the indicated place value. 19. 13.118; hundredth 20. 37.91; tenth 21. 15.992; tenth 22. 173.05; whole number Right Triangles and Trigonometry 515515 �� Key Vocabulary/Vocabulario angle of depression ángulo de depresión angle of elevation ángulo de elevación cosine coseno geometric mean media geométrica sine tangent seno tangente trigonometric ratio razón trigonométrica vector vector Vocabulary Connections To become familiar with some of the vocabulary terms in the chapter, consider the following. You may refer to the chapter, the glossary, or a dictionary if you like. 1. The term angle of elevation includes the word elevation. What does elevate mean in everyday usage? What do you think an angle of elevation might be? 2. A vector is sometimes defined as “a directed line segment.” How can you use this definition to understand this term? 3. The word trigonometric comes from the Greek word trigonon, which means “triangle,” and the suffix metric, which means “measurement.” Based on this, how do you think you might use a trigonometric ratio? Geometry TEKS 8-2 Tech. Lab Les. 8-1 Les. 8-2 Les. 8-3 Les. 8-4 8-4 Geo. Lab Les. 8-5 Les. 8-6 Ext. G.1.B Geometric structure* recognize the historical development of ★ geometric systems... G.2.A Geometric structure* use constructions to explore attributes of geometric figures and to make conjectures about geometric relationships ★ G.5.B Geometric patterns* use … numeric and geometric patterns to make generalizations about geometric properties, including … ratios in similar figures... ★ ★ ★ ★ ★ G.5.D Geometric patterns* identify and apply patterns from right ★ ★ ★ ★ ★ ★ triangles to solve

meaningful problems … G.7.A Dimensionality and the geometry of location* use … two- ★ ★ ★ ★ dimensional coordinate systems … G.8.C Congruence and the geometry of size* … use the ★ ★ ★ Pythagorean Theorem G.10.A Congruence and geometry of size* use congruence transformations to … justify properties of geometric figures including figures represented on a coordinate plane G.11.A Similarity and the geometry of shape* use and extend similarity properties and transformations to explore and justify conjectures about geometric figures G.11.B Similarity and the geometry of shape* use ratios to solve problems involving similar figures G.11.C Similarity and the geometry of shape* develop, apply, and justify triangle similarity relationships, such as right triangle ratios, trigonometric ratios, … ★ ★ ★ ★ ★ ★ ★ ★ ★ ★ ★ ★ ★ ★ ★ ★ ★ * Knowledge and skills are written out completely on pages TX28–TX35. 516 516 Chapter 8 Reading Strategy: Read to Understand As you read a lesson, read with a purpose. Lessons are about one or two specific objectives. These objectives are at the top of the first page of every lesson. Reading with the objectives in mind can help you understand the lesson. Identify similar polygons. Lesson 7-2 Ratios in Similar Polygons Figures that are similar (∼) have the same shape but not necessarily the same size. △1 is similar to △2 (△1 ∼ △2). △1 is not similar to △3 (△1 ≁ △3). Identify the objectives of the lesson. Read through the lesson to find where the objectives are explained. • Can two polygons be both similar and congruent? • In Example 1, the triangles are not oriented the same. How can you tell which angles are congruent and which sides are corresponding? List any questions, problems, or trouble spots you may have. • Similarity is represented by the symbol ∼. Congruence is represented by the symbol ≅. • Similar: same shape but not necessarily the same size Write down any new vocabulary or symbols. Try This Use Lesson 8-1 to complete each of the following. 1. What are the objectives of the lesson? 2. Identify any new vocabulary, formulas, and symbols. 3. Identify any examples that you need clarified. 4. Make a list of questions you

need answered during class. Right Triangles and Trigonometry 517 517 �� 8-1 Similarity in Right Triangles TEKS G.11.C Similarity and the geometry of shape: develop, apply, and justify triangle similarity relationships, such as right triangle ratios.... Objectives Use geometric mean to find segment lengths in right triangles. Apply similarity relationships in right triangles to solve problems. Vocabulary geometric mean Also G.5.B, G.5.D, G.8.A, G.8.C, G.11.A, G.11.B Why learn this? You can use similarity relationships in right triangles to find the height of Big Tex. Big Tex debuted as the official symbol of the State Fair of Texas in 1952. This 6000-pound cowboy wears size 70 boots and a 75-gallon hat. In this lesson, you will learn how to use right triangle relationships to find Big Tex’s height. In a right triangle, an altitude drawn from the vertex of the right angle to the hypotenuse forms two right triangles. Theorem 8-1-1 The altitude to the hypotenuse of a right triangle forms two triangles that are similar to each other and to the original triangle. △ABC ∼ △ACD ∼ △CBD PROOF PROOF Theorem 8-1-1 ̶̶ CD. Given: △ABC is a right triangle with altitude Prove: △ABC ∼ △ACD ∼ △CBD Proof: The right angles in △ABC, △ACD, and △CBD are all congruent. By the Reflexive Property of Congruence, ∠A ≅ ∠A. Therefore △ABC ∼ △ACD by the AA Similarity Theorem. Similarly, ∠B ≅ ∠B, so △ABC ∼ △CBD. By the Transitive Property of Similarity, △ABC ∼ △ACD ∼ △CBD. E X A M P L E 1 Identifying Similar Right Triangles Write a similarity statement comparing the three triangles. Sketch the three right triangles with the angles of the triangles in corresponding positions. By Theorem 8-1-1, △RST ∼ △SPT ∼ △RPS. 1. Write a similarity statement comparing the three triangles. 518 518 Chapter 8 Right Triangles and Trigonometry �� Consider the

proportion a __ x = x __ the same number, and that number is the geometric mean of the extremes. The geometric mean of two positive numbers is the positive square root of their product. So the geometric mean of a and b is the positive number x such that x = √  ab, or x 2 = ab.. In this case, the means of the proportion are b E X A M P L E 2 Finding Geometric Means Find the geometric mean of each pair of numbers. If necessary, give the answer in simplest radical form. A 4 and 9 Let x be the geometric mean. x 2 = (4) (9) = 36 x = 6 Def. of geometric mean Find the positive square root. B 6 and 15 Let x be the geometric mean. x 2 = (6) (15) = 90 x = √  90 = 3 √  10 Def. of geometric mean Find the positive square root. Find the geometric mean of each pair of numbers. If necessary, give the answer in simplest radical form. 2a. 2 and 8 2b. 10 and 30 2c. 8 and 9 You can use Theorem 8-1-1 to write proportions comparing the side lengths of the triangles formed by the altitude to the hypotenuse of a right triangle. All the relationships in red involve geometric means Corollaries Geometric Means COROLLARY EXAMPLE DIAGRAM 8-1-2 The length of the altitude to the hypotenuse of a right triangle is the geometric mean of the lengths of the two segments of the hypotenuse. 8-1-3 The length of a leg of a right triangle is the geometric mean of the lengths of the hypotenuse and the segment of the hypotenuse adjacent to that leg. h 2 = xy a 2 = xc b 2 = yc 8-1 Similarity in Right Triangles 519 519 �� E X A M P L E 3 Finding Side Lengths in Right Triangles Once you’ve found the unknown side lengths, you can use the Pythagorean Theorem to check your answers. Find x, y, and z. x 2 = (2) (10) = 20 x = √  20 = 2 √  5 y 2 = (12) (10) = 120 120 = 2 √  30 y = √ �

� z 2 = (12) (2) = 24 z = √  24 = 2 √  6 x is the geometric mean of 2 and 10. Find the positive square root. y is the geometric mean of 12 and 10. Find the positive square root. z is the geometric mean of 12 and 2. Find the positive square root. 3. Find u, v, and w. E X A M P L E 4 Measurement Application To estimate the height of Big Tex at the State Fair of Texas, Michael steps away from the statue until his line of sight to the top of the statue and his line of sight to the bottom of the statue form a 90° angle. His eyes are 5 ft above the ground, and he is standing 15 ft 3 in. from Big Tex. How tall is Big Tex to the nearest foot? Let x be the height of Big Tex above eye level. 15 ft 3 in. = 15.25 ft Convert 3 in. to 0.25 ft. (15.25) 2 = 5x 15.25 is the geometric mean of 5 and x. x = 46.5125 ≈ 47 Solve for x and round. Big Tex is about 47 + 5, or 52 ft tall. 4. A surveyor positions himself so that his line of sight to the top of a cliff and his line of sight to the bottom form a right angle as shown. What is the height of the cliff to the nearest foot? THINK AND DISCUSS 1. Explain how to find the geometric mean of 7 and 21. 2. GET ORGANIZED Copy and complete the graphic organizer. Label the right triangle and draw the altitude to the hypotenuse. In each box, write a proportion in which the given segment is a geometric mean. 520 520 Chapter 8 Right Triangles and Trigonometry �� 8-1 Exercises Exercises KEYWORD: MG7 8-1 KEYWORD: MG7 Parent GUIDED PRACTICE 1. Vocabulary In the proportion 2 __ 8 = 8 __ 32, which number is the geometric mean of the other two numbers Write a similarity statement comparing the three triangles in each diagram. p. 518 2. 3. 4. 519 Find the geometric mean of each pair of numbers. If

necessary, give the answer in simplest radical form. 5. 2 and 50 8. 9 and 12 6. 4 and 16 9. 16 and 25 7. 1 _ 2 and 8 10. 7 and 11 Find x, y, and z. p. 520 11. 12. 13. 520 14. Measurement To estimate the length of the USS Constitution in Boston harbor, a student locates points T and U as shown. What is RS to the nearest tenth? PRACTICE AND PROBLEM SOLVING Independent Practice Write a similarity statement comparing the three triangles in each diagram. For See Exercises Example 15. 16. 17. 15–17 18–23 24–26 27 1 2 3 4 TEKS TEKS TAKS TAKS Skills Practice p. S18 Application Practice p. S35 Find the geometric mean of each pair of numbers. If necessary, give the answer in simplest radical form. 18. 5 and 45 21. 1 _ 4 and 80 Find x, y, and z. 24. 19. 3 and 15 22. 1.5 and 12 20. 5 and 8 and 27 _ 23. 2 _ 40 3 25. 26. 8-1 Similarity in Right Triangles 521 521 �� 27. Measurement To estimate the height of the Taipei 101 tower, Andrew stands so that his lines of sight to the top and bottom of the tower form a 90° angle. What is the height of the tower to the nearest foot? 28. The geometric mean of two numbers is 8. One of the numbers is 2. Find the other number. 29. The geometric mean of two numbers is 2 √  5. One of the numbers is 6. Find the other number. Use the diagram to complete each equation. 30 33.? 31.? _ u = u _ x 34. (?) 2 = y (x + y) 32. x + y _ v 35. u 2 = (x + y) (?) = v _? Give each answer in simplest radical form. 36. AD = 12, and CD = 8. Find BD. 37. AC = 16, and CD = 5. Find BC. 38. AD = CD = √  2. Find BD. 39. BC = √  5, and AC = √  10. Find CD. 40. Finance

An investment returns 3% one year and 10% the next year. The average rate of return is the geometric mean of the two annual rates. What is the average rate of return for this investment to the nearest tenth of a percent? 41. /////ERROR ANALYSIS///// Two students were asked to find EF. Which solution is incorrect? Explain the error. 42. The altitude to the hypotenuse of a right triangle divides the hypotenuse into segments that are 2 cm long and 5 cm long. Find the length of the altitude to the nearest tenth of a centimeter. 43. Critical Thinking Use the figure to show how Corollary 8-1-3 can be used to derive the Pythagorean Theorem. (Hint: Use the corollary to write expressions for a 2 and b 2. Then add the expressions.) 44. This problem will prepare you for the Multi-Step TAKS Prep on page 542. Before installing a utility pole, a crew must first dig a hole and install the anchor for the guy wire ̶̶ RT, that supports the pole. In the diagram, ̶̶̶ ̶̶̶ WT, RS = 4 ft, and ST = 3 ft. RW ⊥ a. Find the depth of the anchor ̶̶̶ SW to the ̶̶̶ SW ⊥ nearest inch. b. Find the length of the rod ̶̶̶ RW to the nearest inch. 522 522 Chapter 8 Right Triangles and Trigonometry � � � � �� 5 ft91 ft 3 in.ge07sec08l01003a�� 45. Write About It Suppose the rectangle and square have the same area. Explain why s must be the geometric mean of a and b. 46. Write About It Explain why the geometric mean of two perfect squares must be a whole number. 47. Lee is building a skateboard ramp based on the plan shown. Which is closest to the length of the ramp from point X to point Y? 4.9 feet 5.7 feet 8.5 feet 9.4 feet 48. What is the area of △ABC? 18 square meters

36 square meters 39 square meters 78 square meters 49. Which expression represents the length of ̶̶ RS? √  y + 1) CHALLENGE AND EXTEND 50. Algebra An 8-inch-long altitude of a right triangle divides the hypotenuse into two segments. One segment is 4 times as long as the other. What are the lengths of the segments of the hypotenuse? 51. Use similarity in right triangles to find x, y, and z. 52. Prove the following. If the altitude to the hypotenuse of a right triangle bisects the hypotenuse, then the triangle is a 45°-45°-90° right triangle. 53. Multi-Step Find AC and AB to the nearest hundredth. SPIRAL REVIEW Find the x-intercept and y-intercept for each equation. (Previous course) 54. 3y + 4 = 6x 55. x + 4 = 2y 56. 3y - 15 = 15x The leg lengths of a 30°-60°-90° triangle are given. Find the length of the hypotenuse. (Lesson 5-8) 57. 3 and √  27 58. 7 and 7 √  3 59. 2 and 2 √  3 For rhombus ABCD, find each measure, given that m∠DEC = 30y°, m∠EDC = (8y + 15) °, AB = 2x + 8, and BC = 4x. (Lesson 6-4) 61. m∠EDA 60. m∠EDC 62. AB 8-1 Similarity in Right Triangles 523 523 �� 8-2 Explore Trigonometric Ratios In a right triangle, the ratio of two side lengths is known as a trigonometric ratio. Use with Lesson 8-2 Activity TEKS G.2.A Geometric structure: use constructions to explore attributes of geometric figures and to make conjectures about geometric relationships. Also G.5.B, G.9.B, G.11.A KEYWORD: MG7 Lab8 1 Construct three points and label them A, B, and C. Construct rays endpoint A. Move C so that ∠A is an acute angle. �

�� AC with common  AB and 2 Construct point D on  AC. Construct a line  AB. Label the through D perpendicular to intersection of the perpendicular line and  AB as E. 3 Measure ∠A. Measure DE, AE, and AD, the side lengths of △AED. 4 Calculate the ratios DE _ AD, AE _ AD, and DE _. AE Try This 1. Drag D along  AC. What happens to the measure of ∠A as D moves? What postulate or theorem guarantees that the different triangles formed are similar to each other? 2. As you move D, what happens to the values of the three ratios you calculated? Use the properties of similar triangles to explain this result. 3. Move C. What happens to the measure of ∠A? With a new value for m∠A, note the values of the three ratios. What happens to the ratios if you drag D? 4. Move C until DE ___ AD = AE ___ AD. What is the value of DE ___ AE? What is the measure of ∠A? Use the properties of special right triangles to justify this result. 524 524 Chapter 8 Right Triangles and Trigonometry 8-2 Trigonometric Ratios TEKS G.11.C Similarity and the geometry of shape: develop, apply, and justify triangle similarity relationships, such as... trigonometric ratios.... Objectives Find the sine, cosine, and tangent of an acute angle. Use trigonometric ratios to find side lengths in right triangles and to solve real-world problems. Vocabulary trigonometric ratio sine cosine tangent Also G.5.B, G.5.D, G.8.C, G.11.B Who uses this? Contractors use trigonometric ratios to build ramps that meet legal requirements. According to the Americans with Disabilities Act (ADA), the maximum slope allowed for a wheelchair ramp is 1__ 12, which is an angle of about 4.8°. Properties of right triangles help builders construct ramps that meet this requirement. By the AA Similarity Postulate, a right triangle with a given acute angle is similar to every other right triangle with that same acute angle measure. So △ABC ∼ △

DEF ∼△XYZ, and BC ___. These are trigonometric AC ratios. A trigonometric ratio is a ratio of two sides of a right triangle. = EF ___ DF = YZ ___ XZ Trigonometric Ratios DEFINITION The sine of an angle is the ratio of the length of the leg opposite the angle to the length of the hypotenuse. The cosine of an angle is the ratio of the length of the leg adjacent to the angle to the length of the hypotenuse. The tangent of an angle is the ratio of the length of the leg opposite the angle to the length of the leg adjacent to the angle. In trigonometry, the letter of the vertex of the angle is often used to represent the measure of that angle. For example, the sine of ∠A is written as sin A. DIAGRAM SYMBOLS __ _ a = c _ __ b = c opposite leg hypotenuse opposite leg hypotenuse sin A = sin B = cos A = cos B = _ __ b = c _ __ a = c adjacent leg hypotenuse adjacent leg hypotenuse tan A = tan B = _ __ a = b _ __ b = a opposite leg adjacent leg opposite leg adjacent leg E X A M P L E 1 Finding Trigonometric Ratios Write each trigonometric ratio as a fraction and as a decimal rounded to the nearest hundredth. A sin R sin R = 12 _ 13 ≈ 0.92 The sine of an ∠ is opp. leg _ hyp.. 8-2 Trigonometric Ratios 525 525 �� Write each trigonometric ratio as a fraction and as a decimal rounded to the nearest hundredth. B cos R cos R = 5 _ 13 ≈ 0.38 C tan S tan S = 5 _ 12 ≈ 0.42 The cosine of an ∠ is adj. leg _. hyp. The tangent of an ∠ is opp. leg _. adj. leg Write each trigonometric ratio as a fraction and as a decimal rounded to the nearest hundredth. 1a. cos A 1b. tan B 1c. sin B E X A M P L E 2 Finding Trigonometric Ratios in Special Right Triangles Use a special right triangle to write sin 60° as a fraction. Draw and label a 30°-60°-90° △. sin 60° = s √

 3 _ 2s = √  3 _ 2 The sine of an ∠ is opp. leg _ hyp.. 2. Use a special right triangle to write tan 45° as a fraction. E X A M P L E 3 Calculating Trigonometric Ratios Use your calculator to find each trigonometric ratio. Round to the nearest hundredth. A cos 76° B sin 8° C tan 82° Be sure your calculator is in degree mode, not radian mode. cos 76° ≈ 0.24 sin 8° ≈ 0.14 tan 82° ≈ 7.12 Use your calculator to find each trigonometric ratio. Round to the nearest hundredth. 3a. tan 11° 3b. sin 62° 3c. cos 30° The hypotenuse is always the longest side of a right triangle. So the denominator of a sine or cosine ratio is always greater than the numerator. Therefore the sine and cosine of an acute angle are always positive numbers less than 1. Since the tangent of an acute angle is the ratio of the lengths of the legs, it can have any value greater than 0. 526 526 Chapter 8 Right Triangles and Trigonometry �� E X A M P L E 4 Using Trigonometric Ratios to Find Lengths Find each length. Round to the nearest hundredth. A AB ̶̶ AB is adjacent to the given angle, ∠A. You are given BC, which is opposite ∠A. Since the adjacent and opposite legs are involved, use a tangent ratio. Do not round until the final step of your answer. Use the values of the trigonometric ratios provided by your calculator. tan A = opp. leg _ adj. leg = BC _ AB Write a trigonometric ratio. tan 41° = 6.1 _ AB AB = 6.1 _ tan 41° AB ≈ 7.02 in. Substitute the given values. Multiply both sides by AB and divide by tan 41°. Simplify the expression. B MP ̶̶̶ MP is opposite the given angle, ∠N. You are given NP, which is the hypotenuse. Since the opposite side and hypotenuse are involved, use a sine ratio. = MP _ NP sin N = opp. leg _ hyp. sin 20° = MP _ 8.7 8.7 (sin 20°)

= MP Write a trigonometric ratio. Substitute the given values. Multiply both sides by 8.7. MP ≈ 2.98 cm Simplify the expression. C YZ YZ is the hypotenuse. You are given XZ, which is adjacent to the given angle, ∠Z. Since the adjacent side and hypotenuse are involved, use a cosine ratio. cos Z = adj. leg _ hyp. cos 38° = 12.6 _ = XZ _ YZ YZ YZ = 12.6 _ cos 38° Write a trigonometric ratio. Substitute the given values. Multiply both sides by YZ and divide by cos 38°. YZ ≈ 15.99 cm Simplify the expression. Find each length. Round to the nearest hundredth. 4a. DF 4b. ST 4c. BC 4d. JL 8-2 Trigonometric Ratios 527 527 �� E X A M P L E 5 Problem Solving Application A contractor is building a wheelchair ramp for a doorway that is 1.2 ft above the ground. To meet ADA guidelines, the ramp will make an angle of 4.8° with the ground. To the nearest hundredth of a foot, what is the horizontal distance covered by the ramp? Understand the Problem Make a sketch. The answer is BC. Make a Plan ̶̶ BC is the leg adjacent to ∠C. You are given AB, which is the leg opposite ∠C. Since the opposite and adjacent legs are involved, write an equation using the tangent ratio. Solve tan C = AB _ BC tan 4.8° = 1.2 _ BC BC = 1.2 _ tan 4.8° Write a trigonometric ratio. Substitute the given values. Multiply both sides by BC and divide by tan 4.8°. BC ≈ 14.2904 ft Simplify the expression. Look Back The problem asks for BC rounded to the nearest hundredth, so round the length to 14.29. The ramp covers a horizontal distance of 14.29 ft. 5. Find AC, the length of the ramp in Example 5, to the nearest hundredth of a foot. THINK AND DISCUSS 1. Tell how you could use a sine ratio to find AB. 2. Tell how you could use a cos

ine ratio to find AB. 3. GET ORGANIZED Copy and complete the graphic organizer. In each cell, write the meaning of each abbreviation and draw a diagram for each. 528 528 Chapter 8 Right Triangles and Trigonometry ge07sec08l02002aABCAB12��34�� 8-2 Exercises Exercises KEYWORD: MG7 8-2 KEYWORD: MG7 Parent GUIDED PRACTICE Vocabulary Apply the vocabulary from this lesson to answer each question. 1. In △JKL, ∠K is a right angle. Write the sine of ∠J as a ratio of side lengths. 2. In △MNP, ∠M is a right angle. Write the tangent of ∠N as a ratio of side lengths. 525 Write each trigonometric ratio as a fraction and as a decimal rounded to the nearest hundredth. 3. sin C 6. cos C 4. tan A 7. tan C 5. cos A 8. sin Use a special right triangle to write each trigonometric ratio as a fraction. p. 526 9. cos 60° 10. tan 30° 11. sin 45. 526 Use your calculator to find each trigonometric ratio. Round to the nearest hundredth. 12. tan 67° 15. cos 88° 13. sin 23° 16. cos 12° 14. sin 49° 17. tan 9 Find each length. Round to the nearest hundredth. p. 527 18. BC 19. QR 20. KL. 528 21. Architecture A pediment has a pitch of 15°, as shown. If the width of the pediment, WZ, is 56 ft, what is XY to the nearest inch? �� Independent Practice Write each trigonometric ratio as a fraction and as a PRACTICE AND PROBLEM SOLVING For See Exercises Example 22–27 28–30 31–36 37–42 43 1 2 3 4 5 TEKS TEKS TAKS TAKS Skills Practice p. S18 Application Practice p. S35 decimal rounded to the nearest hundredth. 22. cos D 25. cos F 23. tan D 26. sin F 24. tan F 27. sin D Use

a special right triangle to write each trigonometric ratio as a fraction. 28. tan 60° 29. sin 30° 30. cos 45° Use your calculator to find each trigonometric ratio. Round to the nearest hundredth. 31. tan 51° 34. tan 14° 32. sin 80° 35. sin 55° 33. cos 77° 36. cos 48° 8-2 Trigonometric Ratios 529 529 �� Find each length. Round to the nearest hundredth. 37. PQ 38. AC 39. GH 40. 40. XZ 41. KL 42. EF 43. Sports A jump ramp for waterskiing makes an angle of 15° with the surface of the water. The ramp rises 1.58 m above the surface. What is the length of the ramp to the nearest hundredth of a meter? �� Use special right triangles to complete each statement. �� 44. An angle that measures 45. For a 45° angle, the 46. The sine of a? has a tangent of 1. ̶̶̶̶? and ̶̶̶̶? angle is 0.5. ̶̶̶̶? ratios are equal. ̶̶̶̶ Sports The Aquaplex Ski Lake in Austin hosted the 2004 and 2005 Barefoot National Championships. Barefoot skiing began in 1947. The first U.S. national competition was held in Waco in September 1978. 47. The cosine of a 30° angle is equal to the sine of a? angle. ̶̶̶̶ 48. Safety According to the Occupational Safety and Health Administration (OSHA), a ladder that is placed against a wall should make a 75.5° angle with the ground for optimal safety. To the nearest tenth of a foot, what is the maximum height that a 10-ft ladder can safely reach? Find the indicated length in each rectangle. Round to the nearest tenth. 49. BC 50. SU 51. Critical Thinking For what angle measures is the tangent ratio less than 1? greater than 1? Explain. 52. This problem will prepare you for the Multi-Step TAKS Prep on page 542. ̶̶ AB at A utility worker is installing a 25-foot pole ̶̶ AD, will help ̶̶ AC and the foot of a hill. Two guy wires, keep the pole vertical. a. To the nearest inch, how long should b. ̶̶

AD is perpendicular to the hill, which makes an angle of 28° with a horizontal line. To the nearest inch, how long should this guy wire be? ̶̶ AC be? 530 530 Chapter 8 Right Triangles and Trigonometry �� 53. Find the sine of the smaller acute angle in a triangle with side lengths of 3, 4, and 5 inches. History 54. Find the tangent of the greater acute angle in a triangle with side lengths of 7, 24, and 25 centimeters. The Pyramid of Cheops consists of more than 2,000,000 blocks of stone with an average weight of 2.5 tons each. 55. History The Great Pyramid of Cheops in Giza, Egypt, was completed around 2566 B.C.E. Its original height was 482 ft. Each face of the pyramid forms a 52° angle with the ground. To the nearest foot, how long is the base of the pyramid? 56. Measurement Follow these steps to calculate trigonometric ratios. a. Use a centimeter ruler to find AB, BC, and AC. b. Use your measurements from part a to find the sine, cosine, and tangent of ∠A. c. Use a protractor to find m∠A. d. Use a calculator to find the sine, cosine, and tangent of ∠A. e. How do the values in part d compare to the ones you found in part b? 57. Algebra Recall from Algebra I that an identity is an equation that is true for all values of the variables. a. Show that the identity tan A = sin A _ is true when m∠A = 30°. cos A b. Write tan A, sin A, and cos A in terms of a, b, and c. c. Use your results from part b to prove the identity tan A = sin A _. cos A Verify that (sin A) 2 + (cos A) 2 = 1 for each angle measure. 58. m∠A = 45° 61. Multi-Step The equation (sin A) 2 + (cos A) 2 = 1 is known as a 59. m∠ A = 30° 60. m∠A = 60° Pythagorean Identity. a. Write sin A and cos A in terms of a, b, and

c. b. Use your results from part a to prove the identity (sin A) 2 + (cos A) 2 = 1. c. Write About It Why do you think this identity is called a Pythagorean identity? Find the perimeter and area of each triangle. Round to the nearest hundredth. 62. 64. 63. 65. 66. Critical Thinking Draw △ABC with ∠C a right angle. Write sin A and cos B in terms of the side lengths of the triangle. What do you notice? How are ∠A and ∠B related? Make a conjecture based on your observations. 67. Write About It Explain how the tangent of an acute angle changes as the angle measure increases. 8-2 Trigonometric Ratios 531 531 �� 68. Which expression can be used to find AB? 7.1 (sin 25°) 7.1 (cos 25°) 7.1 (sin 65°) 7.1 (tan 65°) 69. A steel cable supports an electrical tower as shown. The cable makes a 65° angle with the ground. The base of the cable is 17 ft from the tower. What is the height of the tower to the nearest foot? 8 feet 15 feet 36 feet 40 feet 70. Which of the following has the same value as sin M? sin N tan M cos N cos M CHALLENGE AND EXTEND Algebra Find the value of x. Then find AB, BC, and AC. Round each to the nearest unit. 71. 72. 73. Multi-Step Prove the identity (tan A) 2 + 1 = 1 _. (cos A) 2 74. A regular pentagon with 1 in. sides is inscribed in a circle. Find the radius of the circle rounded to the nearest hundredth. Each of the three trigonometric ratios has a reciprocal ratio, as defined below. These ratios are cosecant (csc), secant (sec), and cotangent (cot). csc A = 1 _ sin A sec A = 1 _ cot A = 1 _ cos A tan A Find each trigonometric ratio to the nearest hundredth. 75. csc Y 76. sec Z 77. cot Y SPIRAL REVIEW Find three ordered pairs that satisfy each function. (Previous course) 78. f (x) = 3x - 6 79. f (x) = -0.

5x + 10 80. f (x) = x 2 - 4x + 2 Identify the property that justifies each statement. (Lesson 2-5) 81. 82. ̶̶ AB ≅ ̶̶ AB ≅ ̶̶ CD, and ̶̶ AB ̶̶ CD ≅ ̶̶ DE. So ̶̶ AB ≅ ̶̶ DE. 83. If ∠JKM ≅ ∠MLK, then ∠MLK ≅ ∠JKM. Find the geometric mean of each pair of numbers. (Lesson 8-1) 84. 3 and 27 85. 6 and 24 86. 8 and 32 532 532 Chapter 8 Right Triangles and Trigonometry �� Inverse Functions Algebra In Algebra, you learned that a function is a relation in which each element of the domain is paired with exactly one element of the range. If you switch the domain and range of a one-to-one function, you create an inverse function. See Skills Bank page S62 The function y = sin -1 x is the inverse of the function y = sin x. �� If you know the value of a trigonometric ratio, you can use the inverse trigonometric function to find the angle measure. You can do this either with a calculator or by looking at the graph of the function. �� Example Use the graphs above to find the value of x for 1 = sin x. Then write this expression using an inverse trigonometric function. 1 = sin x x = 90° Look at the graph of y = sin x. Find where the graph intersects the line y = 1 and read the corresponding x-coordinate. 90° = sin -1 (1) Switch the x- and y-values. Try This TAKS Grades 9–11 Obj. 2 Use the graphs above to find the value of x for each of the following. Then write each expression using an inverse trigonometric function. 1. 0 = sin x 4. 0 = cos x 2. 1 _ 2 = cos x 5. 0 = tan x 3. 1 = tan x 6. 1 _ 2 = sin x On Track for TAKS 533 533 ��

�� 8-3 Solving Right Triangles TEKS G.11.C Similarity and the geometry of shape: develop, apply, and justify triangle similarity relationships, such as... trigonometric ratios.... Objective Use trigonometric ratios to find angle measures in right triangles and to solve real-world problems. Also G.5.D, G.7.A, G.7.C, G.8.C, G.11.B Why learn this? You can convert the percent grade of a road to an angle measure by solving a right triangle. San Francisco, California, is famous for its steep streets. The steepness of a road is often expressed as a percent grade. Filbert Street, the steepest street in San Francisco, has a 31.5% grade. This means the road rises 31.5 ft over a horizontal distance of 100 ft, which is equivalent to a 17.5° angle. You can use trigonometric ratios to change a percent grade to an angle measure. E X A M P L E 1 Identifying Angles from Trigonometric Ratios Use the trigonometric ratio cos A = 0.6 to determine which angle of the triangle is ∠A. cos A = adj. leg_ hyp. cos ∠1 = 3.6_ 6 cos ∠2 = 4.8_ 6 = 0.6 = 0.8 Cosine is the ratio of the adjacent leg to the hypotenuse. The leg adjacent to ∠1 is 3.6. The hypotenuse is 6. The leg adjacent to ∠2 is 4.8. The hypotenuse is 6. Since cos A = cos ∠1, ∠1 is ∠A. Use the given trigonometric ratio to determine which angle of the triangle is ∠A. 1a. sin A = 8 _ 17 1b. tan A = 1.875 In Lesson 8-2, you learned that sin 30° = 0.5. Conversely, if you know that the sine of an acute angle is 0.5, you can conclude that the angle measures 30°. This is written as sin -1 (0.5) = 30°. If you know the sine, cosine, or tangent of an acute angle measure, you can use the inverse trigonometric functions to find the measure of the angle. Inverse Trigonometric Functions If sin A = x, then sin -1

x = m∠A. If cos A = x, then cos -1 x = m∠A. If tan A = x, then tan -1 x = m∠A. The expression sin -1 x is read “the inverse sine of x.” It does not mean 1 ____. You can think of sin -1 x as “the angle whose sine is x.” sin x 534 534 Chapter 8 Right Triangles and Trigonometry �� E X A M P L E 2 Calculating Angle Measures from Trigonometric Ratios Use your calculator to find each angle measure to the nearest degree. A cos -1 (0.5) B sin -1 (0.45) C tan -1 (3.2) When using your calculator to find the value of an inverse trigonometric expression, you may need to press the [arc], [inv], or [2nd] key. cos -1 (0.5) = 60° sin -1 (0.45) ≈ 27° tan -1 (3.2) ≈ 73° Use your calculator to find each angle measure to the nearest degree. 2a. tan -1 (0.75) 2b. cos -1 (0.05) 2c. sin -1 (0.67) Using given measures to find the unknown angle measures or side lengths of a triangle is known as solving a triangle. To solve a right triangle, you need to know two side lengths or one side length and an acute angle measure. E X A M P L E 3 Solving Right Triangles Find the unknown measures. Round lengths to the nearest hundredth and angle measures to the nearest degree. Method 1: Method 2: By the Pythagorean Theorem, AC 2 = AB 2 + BC 2. = (7.5) 2 + 5 2 = 81.25 So AC = √  81.25 ≈ 9.01. m∠A = tan -1 ( 5 _ ) ≈ 34° 7.5 Since the acute angles of a right triangle are complementary, m∠C ≈ 90° - 34° ≈ 56°. m∠A = tan -1 ( 5 _ ) ≈ 34° 7.5 Since the acute angles of a right triangle are complementary, m∠C ≈ 90° - 34°

≈ 56°. sin A = 5 _ AC, so AC = 5 _. sin A 5 __ ≈ 9.01 ⎤ ⎡ tan -1 ( 5 _ ) ⎥ ⎢ sin ⎦ ⎣ AC ≈ 7.5 3. Find the unknown measures. Round lengths to the nearest hundredth and angle measures to the nearest degree. Solving Right Triangles Rounding can really make a difference! To find AC, I used the Pythagorean Theorem and got 15.62. Then I did it a different way. I used m∠A = tan -1 ( 10 __ find m∠A = 39.8056°, which I rounded to 40°. sin 40° = 10 ___ AC, so AC = 10 _____ ≈ 15.56. sin 40° 12 ) to Kendell Waters Marshall High School The difference in the two answers reminded me not to round values until the last step. 8- 3 Solving Right Triangles 535 535 �� E X A M P L E 4 Solving a Right Triangle in the Coordinate Plane The coordinates of the vertices of △JKL are J (-1, 2), K (-1, -3), and L (3, -3). Find the side lengths to the nearest hundredth and the angle measures to the nearest degree. Step 1 Find the side lengths. Plot points J, K, and L. JK = 5 KL = 4 By the Distance Formula, JL = √  ⎤ ⎦ ⎡ ⎣ 2 + (-3 - 2) 2. 3 - (-1) √  4 2 + (-5) 2 = 16 + 25 = √  41 ≈ 6.40 = √  Step 2 Find the angle measures. m∠K = 90° m∠J = tan -1 ( 4 _ ) ≈ 39° 5 ̶̶ KL are ⊥. ̶̶ JK and ̶̶ KL is opp. ∠J, and ̶̶ JK is adj. to ∠J. m∠L ≈ 90°

- 39° ≈ 51° The acute  of a rt. △ are comp. 4. The coordinates of the vertices of △RST are R (-3, 5), S (4, 5), and T (4, -2). Find the side lengths to the nearest hundredth and the angle measures to the nearest degree. E X A M P L E 5 Travel Application San Francisco’s Lombard Street is known as one of “the crookedest streets in the world.” The road’s eight switchbacks were built in the 1920s to make the steep hill passable by cars. If the hill has a percent grade of 84%, what angle does the hill make with a horizontal line? Round to the nearest degree. 84% = 84 _ 100 Change the percent grade to a fraction. An 84% grade means the hill rises 84 ft for every 100 ft of horizontal distance. Draw a right triangle to represent the hill. ∠A is the angle the hill makes with a horizontal line. m∠A = tan -1 ( 84 _ ) ≈ 40° 100 5. Baldwin St. in Dunedin, New Zealand, is the steepest street in the world. It has a grade of 38%. To the nearest degree, what angle does Baldwin St. make with a horizontal line? 536 536 Chapter 8 Right Triangles and Trigonometry �� THINK AND DISCUSS 1. Describe the steps you would use to solve △RST. 2. Given that cos Z = 0.35, write an equivalent statement using an inverse trigonometric function. 3. GET ORGANIZED Copy and complete the graphic organizer. In each box, write a trigonometric ratio for ∠A. Then write an equivalent statement using an inverse trigonometric function. 8-3 Exercises Exercises GUIDED PRACTICE KEYWORD: MG7 8-3 KEYWORD: MG7 Parent. 534 Use the given trigonometric ratio to determine which angle of the triangle is ∠A. 1. sin A = 4 _ 5 2. tan A = 1 1 _ 3 3. cos A = 0.6 4. cos A = 0.8 5. tan A = 0.75 6. sin A = 0. Use your calculator to find each angle measure to the nearest degree. p. 535 7. tan -1 (2.

1) 10. sin -1 (0.5) 8. cos -1 ( 1 _ ) 11. sin -1 (0.61) 3 9. cos -1 ( 5 _ ) 12. tan -1 (0.09. 535 Multi-Step Find the unknown measures. Round lengths to the nearest hundredth and angle measures to the nearest degree. 13. 14. 15. 536 Multi-Step For each triangle, find the side lengths to the nearest hundredth and the angle measures to the nearest degree. 16. D (4, 1), E (4, -2), F (-2, -2) 17. R (3, 3), S (-2, 3), T (-2, -3) 18. X (4, -6), Y (-3, 1), Z (-3, -6) 19. A (-1, 1), B (1, 1), C (1, 5) 8- 3 Solving Right Triangles 537 537 �� 20. Cycling A hill in the Tour de France p. 536 bike race has a grade of 8%. To the nearest degree, what is the angle that this hill makes with a horizontal line? Independent Practice Use the given trigonometric ratio to determine which angle PRACTICE AND PROBLEM SOLVING For See Exercises Example 21–26 27–32 33–35 36–37 38 1 2 3 4 5 TEKS TEKS TAKS TAKS Skills Practice p. S18 Application Practice p. S35 of the triangle is ∠A. 21. tan A = 5 _ 12 24. sin A = 5 _ 13 22. tan A = 2.4 25. cos A = 12 _ 13 23. sin A = 12 _ 13 26. cos A = 5 _ 13 Use your calculator to find each angle measure to the nearest degree. 27. sin -1 (0.31) 27. 28. tan -1 (1) 30. cos -1 (0.72) 31. tan -1 (1.55) 29. cos -1 (0.8) 32. sin -1 ( 9 _ ) 17 Multi-Step Find the unknown measures. Round lengths to the nearest hundredth and angle measures to the nearest degree. 33. 34

. 35. Multi-Step For each triangle, find the side lengths to the nearest hundredth and the angle measures to the nearest degree. 36. A (2, 0), B (2, -5), C (1, -5) 37. M (3, 2), N (3, -2), P (-1, -2) 38. Building For maximum accessibility, a wheelchair ramp should have a slope between 1 __ 16 and 1 __ 20. What is the range of angle measures that a ramp should make with a horizontal line? Round to the nearest degree. Complete each statement. If necessary, round angle measures to the nearest degree. Round other values to the nearest hundredth.? ≈ 3.5 ̶̶̶̶ 39. tan 42. cos -1 ( 45. Critical Thinking Use trigonometric ratios to explain why the diagonal of? 42° ≈ 0.74 ̶̶̶̶? 60° = 1 _ ̶̶̶̶ 2 40. sin 43. sin -1 (? ≈ 2 _ ̶̶̶̶ 3? ) ≈ 69° ̶̶̶̶? ) ≈ 12° ̶̶̶̶ 41. 44. a square forms a 45° angle with each of the sides. 46. Estimation You can use trigonometry to find angle measures when a protractor is not available. a. Estimate the measure of ∠P. b. Use a centimeter ruler to find RQ and PQ. c. Use your measurements from part b and an inverse trigonometric function to find m∠P to the nearest degree. d. How does your result in part c compare to your estimate in part a? 538 538 Chapter 8 Right Triangles and Trigonometry �� 46. This problem will prepare you for the Multi-Step TAKS Prep on page 542. An electric company wants to install a vertical utility pole at the base of a hill that has an 8% grade. a. To the nearest degree, what angle does the hill make with a horizontal line? b. What is the measure of the angle between the pole and the hill? Round to the nearest degree. c. A utility worker installs a 31-foot guy wire from the top of the pole to the hill. Given that the guy wire is perpendicular to the hill, find the height of the pole to the nearest inch

. The side lengths of a right triangle are given below. Find the measures of the acute angles in the triangle. Round to the nearest degree. 48. 3, 4, 5 49. 5, 12, 13 50. 8, 15, 17 51. What if…? A right triangle has leg lengths of 28 and 45 inches. Suppose the length of the longer leg doubles. What happens to the measure of the acute angle opposite that leg? 52. Fitness As part of off-season training, the Houston Texans football team must sprint up a ramp with a 28% grade. To the nearest degree, what angle does this ramp make with a horizontal line? 53. The coordinates of the vertices of a triangle are A (-1, 0), B (6, 1), and C (0, 3). a. Use the Distance Formula to find AB, BC, and AC. b. Use the Converse of the Pythagorean Theorem to show that △ABC is a right triangle. Identify the right angle. c. Find the measures of the acute angles of △ABC. Round to the nearest degree. Find the indicated measure in each rectangle. Round to the nearest degree. 54. m∠BDC 55. m∠STV Find the indicated measure in each rhombus. Round to the nearest degree. 56. m∠DGF 57. m∠LKN Fitness Running on a treadmill is slightly easier than running outdoors, since you don’t have to overcome wind resistance. Set the treadmill to a 1% grade to match the intensity of an outdoor run. 58. Critical Thinking Without using a calculator, compare the values of tan 60° and tan 70°. Explain your reasoning. The measure of an acute angle formed by a line with slope m and the x-axis can be found by using the expression tan -1 (m). Find the measure of the acute angle that each line makes with the x-axis. Round to the nearest degree. 59. y = 3x + 5 60. y = 2 _ 3 x + 1 61. 5y = 4x + 3 8- 3 Solving Right Triangles 539 539 �� 62. /////ERROR ANALYSIS///// A student was asked to find m∠C. Explain the error in the student’s solution. 63. Write About It

A student claims that you must know the three side lengths of a right triangle before you can use trigonometric ratios to find the measures of the acute angles. Do you agree? Why or why not? ̶̶ DC is an altitude of right △ABC. Use trigonometric ratios to find the missing lengths in the figure. Then use these lengths to verify the three relationships in the Geometric Mean Corollaries from Lesson 8-1. 64. 65. Which expression can be used to find m∠A? tan -1 (0.75) sin -1 ( 3 _ ) 5 cos -1 (0.8) tan -1 ( 4 _ ) 3 66. Which expression is NOT equivalent to cos 60°? 1 _ 2 sin 30° sin 60° _ tan 60° cos -1 ( 1 _ ) 2 67. To the nearest degree, what is the measure of the acute angle formed by Jefferson St. and Madison St.? 27° 31° 59° 63° 68. Gridded Response A highway exit ramp. To the nearest degree, has a slope of 3 __ 20 find the angle that the ramp makes with a horizontal line. CHALLENGE AND EXTEND Find each angle measure. Round to the nearest degree. 69. m∠J 70. m∠A Simply each expression. 71. cos -1 (cos 34°) ⎤ ⎦ ⎡ ⎣ tan -1 (1.5) 72. tan 73. sin ( sin -1 x) 74. A ramp has a 6% grade. The ramp is 40 ft long. Find the vertical distance that the ramp rises. Round your answer to the nearest hundredth. 540 540 Chapter 8 Right Triangles and Trigonometry ��Main St.Jefferson St.Madison St.2.7 mi1.4 mige07se_c08l03007aAB�� 75. Critical Thinking Explain why the expression sin -1 (1.5) does not make sense. 76. If you are given the lengths of two sides of △ABC and the measure of the included angle, you can use the formula 1 __ 2 bc sin A to find the area of the triangle. Derive this formula. (Hint: Draw an altitude from B to ratios to find the length

of this altitude.) ̶̶ AC. Use trigonometric SPIRAL REVIEW The graph shows the amount of rainfall in a city for the first five months of the year. Determine whether each statement is true or false. (Previous course) 77. It rained more in April than it did in January, February, and March combined. 78. The average monthly rainfall for this five- month period was approximately 3.5 inches. 79. The rainfall amount increased at a constant rate each month over the five-month period. Use the diagram to find each value, given that △ABC ≅ △DEF. (Lesson 4-3) 80. x 81. y 82. DF Use your calculator to find each trigonometric ratio. Round to the nearest hundredth. (Lesson 8-2) 83. sin 63° 84. cos 27° 85. tan 64° Using Technology Use a spreadsheet to complete the following. 1. In cells A2 and B2, enter values for the leg lengths of a right triangle. 2. In cell C2, write a formula to calculate c, the length of the hypotenuse. 3. Write a formula to calculate the measure of ∠A in cell D2. Be sure to use the Degrees function so that the answer is given in degrees. Format the value to include no decimal places. 4. Write a formula to calculate the measure of ∠B in cell E2. Again, be sure to use the Degrees function and format the value to include no decimal places. 5. Use your spreadsheet to check your answers for Exercises 48–50. 8- 3 Solving Right Triangles 541 541 �� SECTION 8A Trigonometric Ratios It’s Electrifying! Utility workers install and repair the utility poles and wires that carry electricity from generating stations to consumers. As shown in the figure, a crew of workers ̶̶ AC plans to install a vertical utility pole ̶̶ AB that is and a supporting guy wire perpendicular to the ground. 1. The utility pole is 30 ft tall. The crew finds that DC = 6 ft. What is the distance DB from the pole to the anchor point of the guy wire? 2. How long is the guy wire? Round to the nearest inch. 3

. In the figure, ∠ABD is called the line angle. In order to choose the correct weight of the cable for the guy wire, the crew needs to know the measure of the line angle. Find m∠ABD to the nearest degree. 4. To the nearest degree, what is the measure of the angle formed by the pole and the guy wire? 5. What is the percent grade of the hill on which the crew is working? 542 542 Chapter 8 Right Triangles and Trigonometry �� SECTION 8A Quiz for Lessons 8-1 Through 8-3 8-1 Similarity in Right Triangles Find the geometric mean of each pair of numbers. If necessary, give the answer in simplest radical form. 1. 5 and 12 2. 2.75 and 44 and 15 _ 3. 5 _ 8 2 Find x, y, and z. 4. 5. 6. 7. A land developer needs to know the distance across a pond on a piece of property. What is AB to the nearest tenth of a meter? �� 8-2 Trigonometric Ratios Use a special right triangle to write each trigonometric ratio as a fraction. 8. tan 45° 9. sin 30° 10. cos 30° Use your calculator to find each trigonometric ratio. Round to the nearest hundredth. 11. sin 16° 12. cos 79° 13. tan 27° Find each length. Round to the nearest hundredth. 14. QR 15. AB 16. LM 8-3 Solving Right Triangles Find the unknown measures. Round lengths to the nearest hundredth and angle measures to the nearest degree. 17. 18. 19. 20. The wheelchair ramp at the entrance of the Mission Bay Library has a slope of 1 __ 18. What angle does the ramp make with the sidewalk? Round to the nearest degree. Ready to Go On? 543 543 �� 8-4 Angles of Elevation and Depression TEKS G.11.C Similarity and the geometry of shape:... apply... triangle similarity relationships, such as... trigonometric ratios.... Also G.5.D Objective Solve problems involving angles of elevation and angles of depression. Who uses this? Pilots and air traffic controllers use angles of depression to calculate distances. Vocabulary angle of elevation angle of depression An angle of elevation is the angle formed by

a horizontal line and a line of sight to a point above the line. In the diagram, ∠1 is the angle of elevation from the tower T to the plane P. An angle of depression is the angle formed by a horizontal line and a line of sight to a point below the line. ∠2 is the angle of depression from the plane to the tower. Since horizontal lines are parallel, ∠1 ≅ ∠2 by the Alternate Interior Angles Theorem. Therefore the angle of elevation from one point is congruent to the angle of depression from the other point. E X A M P L E 1 Classifying Angles of Elevation and Depression Classify each angle as an angle of elevation or angle of depression. A ∠3 ∠3 is formed by a horizontal line and a line of sight to a point below the line. It is an angle of depression. B ∠4 ∠4 is formed by a horizontal line and a line of sight to a point above the line. It is an angle of elevation. Use the diagram above to classify each angle as an angle of elevation or angle of depression. 1a. ∠5 1b. ∠6 544 544 Chapter 8 Right Triangles and Trigonometry PT12ge07se_c08l04002aABAngle of depressionAngle of elevation3465ge07se_c08l04006aAB E X A M P L E 2 Finding Distance by Using Angle of Elevation An air traffic controller at an airport sights a plane at an angle of elevation of 41°. The pilot reports that the plane’s altitude is 4000 ft. What is the horizontal distance between the plane and the airport? Round to the nearest foot. Draw a sketch to represent the given information. Let A represent the airport and let P represent the plane. Let x be the horizontal distance between the plane and the airport. tan 41° = 4000 _ x x = 4000 _ tan 41° You are given the side opposite ∠A, and x is the side adjacent to ∠A. So write a tangent ratio. Multiply both sides by x and divide both sides by tan 41°. x ≈ 4601 ft Simplify the expression. 2. What if…? Suppose the plane is at an altitude of 3500 ft and the angle of elevation from the airport to the plane is 29°. What is the horizontal distance between the plane and the airport? Round to the nearest

foot. E X A M P L E 3 Finding Distance by Using Angle of Depression A forest ranger in a 90-foot observation tower sees a fire. The angle of depression to the fire is 7°. What is the horizontal distance between the tower and the fire? Round to the nearest foot. Draw a sketch to represent the given information. Let T represent the top of the tower and let F represent the fire. Let x be the horizontal distance between the tower and the fire. The angle of depression may not be one of the angles in the triangle you are solving. It may be the complement of one of the angles in the triangle. By the Alternate Interior Angles Theorem, m∠F = 7°. tan 7° = 90 _ x Write a tangent ratio. x = 90 _ tan 7° Multiply both sides by x and divide both sides by tan 7°. x ≈ 733 ft Simplify the expression. 3. What if…? Suppose the ranger sees another fire and the angle of depression to the fire is 3°. What is the horizontal distance to this fire? Round to the nearest foot. 8- 4 Angles of Elevation and Depression 545 545 �� E X A M P L E 4 Aviation Application A pilot flying at an altitude of 2.7 km sights two control towers directly in front of her. The angle of depression to the base of one tower is 37°. The angle of depression to the base of the other tower is 58°. What is the distance between the two towers? Round to the nearest tenth of a kilometer. Step 1 Draw a sketch. Let P represent the plane and let A and B represent the two towers. Let x be the distance between the towers. Always make a sketch to help you correctly place the given angle measure. Step 2 Find y. By the Alternate Interior Angles Theorem, m∠CAP = 58°. In △APC, tan 58° = 2.7 _ y. So y = 2.7 _ ≈ 1.6871 km. tan 58° Step 3 Find z. By the Alternate Interior Angles Theorem, m∠CBP = 37°. In △BPC, tan 37° = 2.7 _ z. So z = 2.7 _ ≈ 3.5830 km. tan 37° Step 4 Find x. x = z - y x ≈ 3.5830 - 1.6871 ≈

1.9 km So the two towers are about 1.9 km apart. 4. A pilot flying at an altitude of 12,000 ft sights two airports directly in front of him. The angle of depression to one airport is 78°, and the angle of depression to the second airport is 19°. What is the distance between the two airports? Round to the nearest foot. THINK AND DISCUSS 1. Explain what happens to the angle of elevation from your eye to the top of a skyscraper as you walk toward the skyscraper. 2. GET ORGANIZED Copy and complete the graphic organizer below. In each box, write a definition or make a sketch. 546 546 Chapter 8 Right Triangles and Trigonometry �� 8-4 Exercises Exercises KEYWORD: MG7 8-4 KEYWORD: MG7 Parent GUIDED PRACTICE Vocabulary Apply the vocabulary from this lesson to answer each question. 1. An angle of? is measured from a horizontal line to a point above that line. ̶̶̶̶ (elevation or depression) 2. An angle of? is measured from a horizontal line to a point below that line. ̶̶̶̶ (elevation or depression. 544 Classify each angle as an angle of elevation or angle of depression. 3. ∠1 4. ∠2 5. ∠3 6. ∠. Measurement When the angle of elevation to p. 545 the sun is 37°, a flagpole casts a shadow that is 24.2 ft long. What is the height of the flagpole to the nearest foot. 545 8. Aviation The pilot of a traffic helicopter sights an accident at an angle of depression of 18°. The helicopter’s altitude is 1560 ft. What is the horizontal distance from the helicopter to the accident? Round to the nearest foot. 546 9. Surveying From the top of a canyon, the angle of depression to the far side of the river is 58°, and the angle of depression to the near side of the river is 74°. The depth of the canyon is 191 m. What is the width of the river at the bottom of the canyon? Round to the nearest tenth of a meter. �� Independent Practice For See Exercises Example 10–13 14 15 16 1 2 3 4

TEKS TEKS TAKS TAKS Skills Practice p. S19 Application Practice p. S35 PRACTICE AND PROBLEM SOLVING Classify each angle as an angle of elevation or angle of depression. 10. ∠1 11. ∠2 12. ∠3 13. ∠4 14. Geology To measure the height of a rock formation, a surveyor places her transit 100 m from its base and focuses the transit on the top of the formation. The angle of elevation is 67°. The transit is 1.5 m above the ground. What is the height of the rock formation? Round to the nearest meter. 8- 4 Angles of Elevation and Depression 547 547 1234ge07se_c08l04008a AB1423ge07sec08l04007a�� Space Shuttle Johnson Space Center, in Houston, is home to the Mission Control Center, the base of operations for all space shuttle missions. 15. Forestry A forest ranger in a 120 ft observation tower sees a fire. The angle of 15. depression to the fire is 3.5°. What is the horizontal distance between the tower and the fire? Round to the nearest foot. 16. Space Shuttle Marion is observing the launch of a space shuttle from the command center. When she first sees the shuttle, the angle of elevation to it is 16°. Later, the angle of elevation is 74°. If the command center is 1 mi from the launch pad, how far did the shuttle travel while Marion was watching? Round to the nearest tenth of a mile. Tell whether each statement is true or false. If false, explain why. 17. The angle of elevation from your eye to the top of a tree increases as you walk toward the tree. 18. If you stand at street level, the angle of elevation to a building’s tenth-story window is greater than the angle of elevation to one of its ninth-story windows. 19. As you watch a plane fly above you, the angle of elevation to the plane gets closer to 0° as the plane approaches the point directly overhead. 20. An angle of depression can never be more than 90°. � � � � Use the diagram for Exercises 21 and 22. 21. Which angles are not angles of elevation or angles of depression? 22. The angle of depression from the helicopter to the car is 30°. Find m∠1, m∠2, m∠

3, and m∠4. 23. Critical Thinking Describe a situation in which the angle of depression to an object is decreasing. 24. An observer in a hot-air balloon sights a building that is 50 m from the balloon’s launch point. The balloon has risen 165 m. What is the angle of depression from the balloon to the building? Round to the nearest degree. 25. Multi-Step A surveyor finds that the angle of elevation to the top of a 1000 ft tower is 67°. a. To the nearest foot, how far is the surveyor from the base of the tower? b. How far back would the surveyor have to move so that the angle of elevation to the top of the tower is 55°? Round to the nearest foot. 26. Write About It Two students are using shadows to calculate the height of a pole. One says that it will be easier if they wait until the angle of elevation to the sun is exactly 45°. Explain why the student made this suggestion. 27. This problem will prepare you for the Multi-Step TAKS Prep on page 568. The pilot of a rescue helicopter is flying over the ocean at an altitude of 1250 ft. The pilot sees a life raft at an angle of depression of 31°. a. What is the horizontal distance from the helicopter to the life raft, rounded to the nearest foot? b. The helicopter travels at 150 ft/s. To the nearest second, how long will it take until the helicopter is directly over the raft? 548 548 Chapter 8 Right Triangles and Trigonometry 16º74ºge07sec08l04003_A1 mi �� 28. Mai is flying a plane at an altitude of 1600 ft. She sights a stadium at an angle of depression of 35°. What is Mai’s approximate horizontal distance from the stadium? 676 feet 1120 feet 1450 feet 2285 feet 29. Jeff finds that an office building casts a shadow that is 93 ft long when the angle of elevation to the sun is 60°. What is the height of the building? 54 feet 81 feet 107 feet 161 feet 30. Short Response Jim is rafting down a river that runs through a canyon. He sees a trail marker ahead at the top of the canyon and estimates the angle of elevation from the raft to the marker as 45°. Draw a sketch to represent the situation. Explain what happens to the angle of elevation as Jim moves closer to the marker. CH

ALLENGE AND EXTEND 31. Susan and Jorge stand 38 m apart. From Susan’s position, the angle of elevation to the top of Big Ben is 65°. From Jorge’s position, the angle of elevation to the top of Big Ben is 49.5°. To the nearest meter, how tall is Big Ben? �� 32. A plane is flying at a constant altitude of 14,000 ft and a constant speed of 500 mi/h. The angle of depression from the plane to a lake is 6°. To the nearest minute, how much time will pass before the plane is directly over the lake? 33. A skyscraper stands between two school buildings. The two schools are 10 mi apart. From school A, the angle of elevation to the top of the skyscraper is 5°. From school B, the angle of elevation is 2°. What is the height of the skyscraper to the nearest foot? 34. Katie and Kim are attending a theater performance. Katie’s seat is at floor level. She looks down at an angle of 18° to see the orchestra pit. Kim’s seat is in the balcony directly above Katie. Kim looks down at an angle of 42° to see the pit. The horizontal distance from Katie’s seat to the pit is 46 ft. What is the vertical distance between Katie’s seat and Kim’s seat? Round to the nearest inch. SPIRAL REVIEW 35. Emma and her mother jog along a mile-long circular path in opposite directions. They begin at the same place and time. Emma jogs at a pace of 4 mi/h, and her mother runs at 6 mi/h. In how many minutes will they meet? (Previous course) 36. Greg bought a shirt that was discounted 30%. He used a coupon for an additional 15% discount. What was the original price of the shirt if Greg paid $17.85? (Previous course) Tell which special parallelograms have each given property. (Lesson 6-5) 37. The diagonals are perpendicular. 38. The diagonals are congruent. 39. The diagonals bisect each other. 40. Opposite angles are congruent. Find each length. (Lesson 8-1) 41. x 42. y 43. z 8- 4 Angles of Elevation and Depression 549 549 �� 8-4 Indirect Measurement

Using Trigonometry A clinometer is a surveying tool that is used to measure angles of elevation and angles of depression. In this lab, you will make a simple clinometer and use it to find indirect measurements. Choose a tall object, such as a flagpole or tree, whose height you will measure. TEKS G.11.C Similarity and the geometry of shape:... apply... triangle similarity relationships, such as... trigonometric ratios.... Also G.5.D, G.11.B Use with Lesson 8-4 Activity 1 Follow these instructions to make a clinometer. a. Tie a washer or paper clip to the end of a 6-inch string. b. Tape the string’s other end to the midpoint of the straight edge of a protractor. c. Tape a straw along the straight edge of the protractor. 2 Stand back from the object you want to measure. Use a tape measure to measure and record the distance from your feet to the base of the object. Also measure the height of your eyes above the ground. 3 Hold the clinometer steady and look through the straw to sight the top of the object you are measuring. When the string stops moving, pinch it against the protractor and record the acute angle measure. Try This 1. How is the angle reading from the clinometer related to the angle of elevation from your eye to the top of the object you are measuring? 2. Draw and label a diagram showing the object and the measurements you made. Then use trigonometric ratios to find the height of the object. 3. Repeat the activity, measuring the angle of elevation to the object from a different distance. How does your result compare to the previous one? 4. Describe possible measurement errors that can be made in the activity. 5. Explain why this method of indirect measurement is useful in real-world situations. 550 550 Chapter 8 Right Triangles and Trigonometry 8-5 Law of Sines and Law of Cosines TEKS G.11.C Similarity and the geometry of shape: develop, apply, and justify triangle similarity relationships, such as... trigonometric ratios.... Also G.5.B, G.5.D, G.7.A, G.11.A Objective Use the Law of Sines and the Law of Cosines to solve triangles. Who uses this? Engineers can use the Law of Sines and the Law of Cosines to solve construction problems. Since its completion in 1370

, engineers have proposed many solutions for lessening the tilt of the Leaning Tower of Pisa. The tower does not form a right angle with the ground, so the engineers have to work with triangles that are not right triangles. In this lesson, you will learn to solve any triangle. To do so, you will need to calculate trigonometric ratios for angle measures up to 180°. You can use a calculator to find these values. E X A M P L E 1 Finding Trigonometric Ratios for Obtuse Angles You will learn more about trigonometric ratios of angle measures greater than or equal to 90° in the Chapter Extension. Use a calculator to find each trigonometric ratio. Round to the nearest hundredth. A sin 135° B tan 98° C cos 108° sin 135° ≈ 0.71 tan 98° ≈ -7.12 cos 108° ≈ -0.31 Use a calculator to find each trigonometric ratio. Round to the nearest hundredth. 1a. tan 175° 1c. sin 160° 1b. cos 92° You can use the altitude of a triangle to find a relationship between the triangle’s side lengths. In △ABC, let h represent the length of the ̶̶ AB. altitude from C to, and sin B = h _ a. From the diagram, sin A = h _ b By solving for h, you find that h = b sin A and h = a sin B. So b sin A = a sin B, and sin A _ a You can use another altitude to show that these ratios equal sin C _. c = sin B _. b 8-5 Law of Sines and Law of Cosines 551 551 �� Theorem 8-5-1 The Law of Sines For any △ABC with side lengths a, b, and c, sin A _ a = sin C _ c = sin B _. b You can use the Law of Sines to solve a triangle if you are given • two angle measures and any side length (ASA or AAS) or • two side lengths and a non-included angle measure (SSA). E X A M P L E 2 Using the Law of Sines Find each measure. Round lengths to the nearest tenth and angle measures to the nearest degree. A DF sin D _ EF sin 105° _ 18 = sin E _ DF = sin 32° _ DF Law of Sines Substitute the

given values. DF sin 105° = 18 sin 32° Cross Products Property In a proportion with three parts, you can use any of the two parts together. DF = 18 sin 32° _ ≈ 9.9 sin 105° B m∠S sin T _ RS sin 75° _ 7 = sin S _ RT = sin S _ 5 sin S = 5 sin 75° _ m∠S ≈ sin -1 ( 5 sin 75° _ 7 7 ) ≈ 44° Divide both sides by sin 105°. Law of Sines Substitute the given values. Multiply both sides by 5. Use the inverse sine function to find m∠S. Find each measure. Round lengths to the nearest tenth and angle measures to the nearest degree. 2a. NP 2b. m∠L 2c. m∠X 2d. AC The Law of Sines cannot be used to solve every triangle. If you know two side lengths and the included angle measure or if you know all three side lengths, you cannot use the Law of Sines. Instead, you can apply the Law of Cosines. 552 552 Chapter 8 Right Triangles and Trigonometry �� Theorem 8-5-2 The Law of Cosines For any △ABC with side lengths a, b, and c: a 2 = b 2 + c 2 - 2bc cos A b 2 = a 2 + c 2 - 2ac cos B c 2 = a 2 + b 2 - 2ab cos C The angle referenced in the Law of Cosines is across the equal sign from its corresponding side. You will prove one case of the Law of Cosines in Exercise 57. You can use the Law of Cosines to solve a triangle if you are given • two side lengths and the included angle measure (SAS) or • three side lengths (SSS). E X A M P L E 3 Using the Law of Cosines Find each measure. Round lengths to the nearest tenth and angle measures to the nearest degree. A BC BC 2 = AB 2 + AC 2 - 2 (AB) (AC) cos A = 14 2 + 9 2 - 2 (14 ) (9 ) cos 62° Law of Cosines Substitute the given BC 2 ≈ 158.6932 BC ≈ 12.6 values. Simplify. Find the square root of both sides. B m∠R ST

2 = RS 2 + RT 2 - 2 (RS) (RT) cos R Law of 9 2 = 4 2 + 7 2 -2 (4 ) (7 ) cos R 81 = 65 - 56 cos R 16 = -56 cos R cos R = - 16 _ 56 m∠R = cos -1 (- 16 _ 56 ) ≈ 107° Cosines Substitute the given values. Simplify. Subtract 65 from both sides. Solve for cos R. Use the inverse cosine function to find m∠R. Find each measure. Round lengths to the nearest tenth and angle measures to the nearest degree. 3a. DE 3b. m∠K 3c. YZ 3d. m∠R 8-5 Law of Sines and Law of Cosines 553 553 �� E X A M P L E 4 Engineering Application The Leaning Tower of Pisa is 56 m tall. In 1999, the tower made a 100° angle with the ground. To stabilize the tower, an engineer considered attaching a cable from the top of the tower to a point that is 40 m from the base. How long would the cable be, and what angle would it make with the ground? Round the length to the nearest tenth and the angle measure to the nearest degree. Step 1 Find the length of the cable. AC 2 = AB 2 + BC 2 - 2 (AB) (BC) cos B = 40 2 + 56 2 - 2 (40 ) (56 ) cos 100° AC 2 ≈ 5513.9438 AC ≈ 74.3 m Law of Cosines Substitute the given values. Simplify. Find the square root of both sides. Do not round your answer until the final step of the computation. If a problem has multiple steps, store the calculated answers to each part in your calculator. Step 2 Find the measure of the angle the cable would make with the ground. sin A _ = sin B _ BC AC sin A _ ≈ sin 100° _ 56 74.2559 sin A ≈ 56 sin 100° _ m∠A ≈ sin -1 ( 56 sin 100° _ 74.2559 74.2559 ) ≈ 48° Law of Sines Substitute the calculated value for AC. Multiply both sides by 56. Use the inverse sine function to find m∠A. 4. What if…? Another engineer suggested

using a cable attached from the top of the tower to a point 31 m from the base. How long would this cable be, and what angle would it make with the ground? Round the length to the nearest tenth and the angle measure to the nearest degree. THINK AND DISCUSS 1. Tell what additional information, if any, is needed to find BC using the Law of Sines. 2. GET ORGANIZED Copy and complete the graphic organizer. Tell which law you would use to solve each given triangle and then draw an example. 554 554 Chapter 8 Right Triangles and Trigonometry 100º56 m40 mABCge07sec08l05002a�� 8-5 Exercises Exercises KEYWORD: MG7 8-5 KEYWORD: MG7 Parent GUIDED PRACTICE Use a calculator to find each trigonometric ratio. Round to the nearest hundredth. p. 551 1. sin 100° 4. tan 141° 7. sin 147° 2. cos 167° 5. cos 133° 8. tan 164° 3. tan 92° 6. sin 150° 9. cos 156. 552 Find each measure. Round lengths to the nearest tenth and angle measures to the nearest degree. 10. RT 11. m∠B 12. m∠ 13. m∠Q 14. MN 15. AB p. 553. 554 16. Carpentry A carpenter makes a triangular frame by joining three pieces of wood that are 20 cm, 24 cm, and 30 cm long. What are the measures of the angles of the triangle? Round to the nearest degree. Independent Practice Use a calculator to find each trigonometric ratio. Round to the nearest hundredth. PRACTICE AND PROBLEM SOLVING For See Exercises Example 17–25 26–31 32–37 38 1 2 3 4 TEKS TEKS TAKS TAKS Skills Practice p. S19 Application Practice p. S35 17. cos 95° 20. sin 132° 23. tan 139° 18. tan 178° 21. sin 98° 24. cos 145° 19. tan 118° 22. cos 124° 25. sin 128° Find each measure. Round lengths to the nearest tenth and

angle measures to the nearest degree. 26. m∠C 27. PR 28. JL 29. EF 30. m∠J 31. m∠X 8-5 Law of Sines and Law of Cosines 555 555 �� Surveying Many modern surveys are done with GPS (Global Positioning System) technology. GPS uses orbiting satellites as reference points from which other locations are established. Find each measure. Round lengths to the nearest tenth and angle measures to the nearest degree. 32. AB 33. m∠Z 34. m∠R 35. EF 36. LM 37. m∠G 38. Surveying To find the distance across a lake, a surveyor locates points A, B, and C as shown. What is AB to the nearest tenth of a meter, and what is m∠B to the nearest degree? Use the figure for Exercises 39–42. Round lengths to the nearest tenth and angle measures to the nearest degree. 39. m∠A = 74°, m∠B = 22°, and b = 3.2 cm. Find a. 40. m∠C = 100°, a = 9.5 in., and b = 7.1 in. Find c. 41. a = 2.2 m, b = 3.1 m, and c = 4 m. Find m∠B. 42. a = 10.3 cm, c = 8.4 cm, and m∠A = 45°. Find m∠C. 43. Critical Thinking Suppose you are given the three angle measures of a triangle. Can you use the Law of Sines or the Law of Cosines to find the lengths of the sides? Why or why not? 44. What if…? What does the Law of Cosines simplify to when the given angle is a right angle? 45. Orienteering The map of a beginning orienteering course is shown at right. To the nearest degree, at what angle should a team turn in order to go from the first checkpoint to the second checkpoint? Multi-Step Find the perimeter of each triangle. Round to the nearest tenth. 46. 47. 48. 49. The ambiguous case of the Law of Sines occurs when you are given an acute angle measure and when the side opposite this angle is shorter than the

other given side. In this case, there are two possible triangles. Find two possible values for m∠C to the nearest degree. (Hint: The inverse sine function on your calculator gives you only acute angle measures. Consider this angle and its supplement.) 556 556 Chapter 8 Right Triangles and Trigonometry 6 km4 km3 kmSecondcheckpointFirstcheckpointStart?ge07se_c08L05003atopo mapGeometry 2007 SEHolt Rinehart WinstonKaren Minot(415)883-6560�� 50. This problem will prepare you for the Multi-Step TAKS Prep on page 568. Rescue teams at two heliports, A and B, receive word of a fire at F. a. What is m∠AFB? b. To the nearest mile, what are the distances from each heliport to the fire? c. If a helicopter travels 150 mi/h, how much time is saved by sending a helicopter from A rather than B? Identify whether you would use the Law of Sines or Law of Cosines as the first step when solving the given triangle. 51. 52. 53. 54. The coordinates of the vertices of △RST are R (0, 3), S (3, 1), and T (-3, -1). a. Find RS, ST, and RT. b. Which angle of △RST is the largest? Why? c. Find the measure of the largest angle in △RST to the nearest degree. 55. Art Jessika is creating a pattern for a piece of stained glass. Find BC, AB, and m∠ABC. Round lengths to the nearest hundredth and angle measures to the nearest degree. 56. /////ERROR ANALYSIS///// Two students were asked to find x in △DEF. Which solution is incorrect? Explain the error. 57. Complete the proof of the Law of Cosines for the case when △ABC is an acute triangle. Given: △ABC is acute with side lengths a, b, and c. Prove: a 2 = b 2 + c 2 - 2bc cos A Proof: Draw the altitude from C to ̶̶ AB. Let h be the length of this

altitude. ̶̶ AB into segments of lengths x and y. By the Pythagorean Theorem, It divides a 2 = a. to get c. expression for b 2 to get d. Therefore a 2 = b 2 + c 2 - 2bc cos A by f.?, and b. ̶̶̶̶?. Rearrange the terms to get - 2cx. Substitute the ̶̶̶̶? = h 2 + x 2. Substitute y = c - x into the first equation ̶̶̶̶?. From the diagram, cos A = x __. So x = e. ̶̶̶̶ b?. ̶̶̶̶?. ̶̶̶̶ 58. Write About It Can you use the Law of Sines to solve △EFG? Explain why or why not. 8-5 Law of Sines and Law of Cosines 557 557 �� 59. Which of these is closest to the length of ̶̶ AB? 5.5 centimeters 7.5 centimeters 14.4 centimeters 22.2 centimeters 60. Which set of given information makes it possible to find x using the Law of Sines? m∠T = 38°, RS = 8.1, ST = 15.3 RS = 4, m∠S = 40°, ST = 9 m∠R = 92°, m∠S = 34°, ST = 7 m∠R = 105°, m∠S = 44°, m∠T = 31° 61. A surveyor finds that the face of a pyramid makes a 135° angle with the ground. From a point 100 m from the base of the pyramid, the angle of elevation to the top is 25°. ̶̶ XY? How long is the face of the pyramid, 48 meters 81 meters 124 meters 207 meters CHALLENGE AND EXTEND 62. Multi-Step Three circular disks are placed next to each other as shown. The disks

have radii of 2 cm, 3 cm, and 4 cm. The centers of the disks form △ABC. Find m∠ACB to the nearest degree. 63. Line ℓ passes through points (-1, 1) and (1, 3). Line m passes through points (-1, 1) and (3, 2). Find the measure of the acute angle formed by ℓ and m to the nearest degree. 64. Navigation The port of Bonner is 5 mi due south of the port of Alston. A boat leaves the port of Alston at a bearing of N 32° E and travels at a constant speed of 6 mi/h. After 45 minutes, how far is the boat from the port of Bonner? Round to the nearest tenth of a mile. SPIRAL REVIEW Write a rule for the nth term in each sequence. (Previous course) 65. 3, 6, 9, 12, 15, … 66. 3, 5, 7, 9, 11, … 67. 4, 6, 8, 10, 12, … State the theorem or postulate that justifies each statement. (Lesson 3-2) 68. ∠1 ≅ ∠8 69. ∠4 ≅ ∠5 70. m∠4 + m∠6 = 180° 71. ∠2 ≅ ∠7 Use the given trigonometric ratio to determine which angle of the triangle is ∠A. (Lesson 8-3) 73. sin A = 15 _ 72. cos A = 15 _ 17 17 74. tan A = 1.875 558 558 Chapter 8 Right Triangles and Trigonometry ��YX135°25°100 mge07se_c08l05005aA2 cm4 cm3 cmBCge07se_c08l05006aaAB�� 8-6 Vectors TEKS G.7.A Dimensionality and the geometry of location: use... two-dimensional coordinate systems to represent... line segments, and figures. Objectives Find the magnitude and direction of a vector. Use vectors and vector addition to solve realworld problems. Who uses this? By using vectors, a kayaker can take water currents into account when planning a course. (See Example 5.) The speed and direction an object moves can be represented by a vector. A vector is a quantity that has both length and direction. You can

think of a vector as a directed line segment. The vector below may be named  AB or  v. Vocabulary vector component form magnitude direction equal vectors parallel vectors resultant vector Also G.1.B, G.7.C, G.11.C A vector can also be named using component form. The component form 〈x, y〉 of a vector lists the horizontal and vertical change from the initial point to   CD is 〈2, 3〉. the terminal point. The component form of E X A M P L E 1 Writing Vectors in Component Form Write each vector in component form. A B   EF The horizontal change from E to F is 4 units. The vertical change from E to F is -3 units. So the component form of  EF is 〈4, -3〉.   PQ with P (7, -5) and Q (4, 3)  PQ = 〈〉 Subtract the coordinates of the initial point  PQ = 〈4 - 7, 3 - (-5) 〉  PQ = 〈-3, 8〉 from the coordinates of the terminal point. Substitute the coordinates of the given points. Simplify. Write each vector in component form. 1a.  u 1b. the vector with initial point L (-1, 1) and terminal point M (6, 2) 8-6 Vectors 559 559 �� The magnitude of a vector is its length. The magnitude of a vector is written  AB ⎟ or ⎜ ⎜  v ⎟. When a vector is used to represent speed in a given direction, the magnitude of the vector equals the speed. For example, if a vector represents the course a kayaker paddles, the magnitude of the vector is the kayaker’s speed. E

X A M P L E 2 Finding the Magnitude of a Vector Draw the vector 〈4, -2〉 on a coordinate plane. Find its magnitude to the nearest tenth. Step 1 Draw the vector on a coordinate plane. Use the origin as the initial point. Then (4, -2) is the terminal point. Step 2 Find the magnitude. Use the Distance Formula. ⎜〈4, -2〉⎟ = √  (4 - 0) 2 + (-2 - 0) 2 = √  20 ≈ 4.5 2. Draw the vector 〈-3, 1〉 on a coordinate plane. Find its magnitude to the nearest tenth. The direction of a vector is the angle that it makes with a horizontal line. This angle is measured counterclockwise from the positive x-axis.  AB is 60°. The direction of See Lesson 4-5, page 252, to review bearings. The direction of a vector can also be given as a bearing relative to the compass directions north, south, east, and west. of N 30° E.  AB has a bearing E X A M P L E 3 Finding the Direction of a Vector A wind velocity is given by the vector 〈2, 5〉. Draw the vector on a coordinate plane. Find the direction of the vector to the nearest degree. Step 1 Draw the vector on a coordinate plane. Use the origin as the initial point. Step 2 Find the direction. Draw right triangle ABC as shown. ∠A is the angle formed by the vector and the x-axis, and tan A = 5 _ 2. So m∠A = tan -1 ( 5 _ ) ≈ 68°. 2 3. The force exerted by a tugboat is given by the vector 〈7, 3〉. Draw the vector on a coordinate plane. Find the direction of the vector to the nearest degree. 560 560 Chapter 8 Right Triangles and Trigonometry �� Two vectors are equal vectors if they have the same  v.  u = magnitude and the same direction. For example, Equal vectors do not have to have the same

initial point and terminal point. Two vectors are parallel vectors if they have the same direction or if they have opposite directions. They may have different magnitudes. For example, vectors are always parallel vectors.  x. Equal  w ǁ  BA  AB ≠ Note that since these vectors do not have the same direction. E X A M P L E 4 Identifying Equal and Parallel Vectors Identify each of the following. A equal vectors  AB =  GH Identify vectors with the same magnitude and direction. B parallel vectors  AB ǁ  GH and  CD ǁ  EF Identify vectors with the same or opposite directions. Identify each of the following. 4a. equal vectors 4b. parallel vectors The resultant vector is the vector that represents the sum of two given vectors. To add two vectors geometrically, you can use the head-to-tail method or the parallelogram method. Vector Addition METHOD EXAMPLE Head-to-Tail Method Place the initial point (tail) of the second vector on the terminal point (head) of the first vector. The resultant is the vector that joins the initial point of the first vector to the terminal point of the second vector. Parallelogram Method Use the same initial point for both of the given vectors. Create a parallelogram by adding a copy of each vector at the terminal point (head) of the other vector. The resultant vector is a diagonal of the parallelogram formed. 8-6 Vectors 561 561 �� To add vectors numerically, add their components. If  v = 〈 x 2, y 2 〉, then  〉.  〉 and E X A M P L

E 5 Sports Application A kayaker leaves shore at a bearing of N 55° E and paddles at a constant speed of 3 mi/h. There is a 1 mi/h current moving due east. What are the kayak’s actual speed and direction? Round the speed to the nearest tenth and the direction to the nearest degree. Step 1 Sketch vectors for the kayaker and the current. Component form gives the horizontal and vertical change from the initial point to the terminal point of the vector. Step 2 Write the vector for the kayaker in component form. The kayaker’s vector has a magnitude of 3 mi/h and makes an angle of 35° with the x-axis., so x = 3 cos 35° ≈ 2.5. cos 35° = x _ 3 y _, so y = 3 sin 35° ≈ 1.7. sin 35° = 3 The kayaker’s vector is 〈2.5, 1.7〉. Step 3 Write the vector for the current in component form. Since the current moves 1 mi/h in the direction of the x-axis, it has a horizontal component of 1 and a vertical component of 0. So its vector is 〈1, 0〉. Step 4 Find and sketch the resultant vector AB. Add the components of the kayaker’s vector and the current’s vector. 〈2.5, 1.7〉 + 〈1, 0〉 = 〈3.5, 1.7〉 The resultant vector in component form is 〈3.5, 1.7〉. Step 5 Find the magnitude and direction of the resultant vector. The magnitude of the resultant vector is the kayak’s actual speed. ⎜〈3.5, 1.7〉⎟ = √ (3.5 - 0)2 + (1.7 - 0)2 ≈ 3.9 mi/h The angle measure formed by the resultant vector gives the kayak’s actual direction. tan A = 1.7_ 3.5 3.5) ≈ 26°, or N 64° E., so A = tan -1(1.7

_ 5. What if…? Suppose the kayaker in Example 5 instead paddles at 4 mi/h at a bearing of N 20° E. What are the kayak’s actual speed and direction? Round the speed to the nearest tenth and the direction to the nearest degree. 562 562 Chapter 8 Right Triangles and Trigonometry ��〈��〉�� THINK AND DISCUSS 1. Explain why the segment with endpoints (0, 0) and (1, 4) is not a vector. 2. Assume you are given a vector in component form. Other than the Distance Formula, what theorem can you use to find the vector’s magnitude? 3. Describe how to add two vectors numerically. 4. GET ORGANIZED Copy and complete the graphic organizer. 8-6 Exercises Exercises KEYWORD: MG7 8-6 KEYWORD: MG7 Parent GUIDED PRACTICE Vocabulary Apply the vocabulary from this lesson to answer each question. 1. 2.? vectors have the same magnitude and direction. (equal, parallel, or resultant) ̶̶̶̶? vectors have the same or opposite directions. (equal, parallel, or resultant) ̶̶̶̶ 3. The? of a vector indicates the vector’s size. (magnitude or direction) ̶̶̶̶ Write each vector in component form. p. 559 4.  AC with A (1, 2) and C (6, 5) 5. the vector with initial point M (-4, 5) and terminal point N (4, -3) 6.  PQ Draw each vector on a coordinate plane. Find its magnitude to the nearest tenth. p. 560 7. 〈1, 4〉 8. 〈-3, -2〉 9. 〈5, -3. 560 Draw each vector on a coordinate plane. Find the direction of the vector to the nearest degree. 10. A river’s current is given by the vector 〈4, 6〉. 11. The velocity of a plane is given by the vector 〈5, 1〉. 12. The path of a hiker is given by the vector 〈6

, 3〉. Identify each of the following. p. 561 13. equal vectors in diagram 1 14. parallel vectors in diagram 1 15. equal vectors in diagram 2 16. parallel vectors in diagram 2 8-6 Vectors 563 563 ��. 562 17. Recreation To reach a campsite, a hiker first walks for 2 mi at a bearing of N 40° E. Then he walks 3 mi due east. What are the magnitude and direction of his hike from his starting point to the campsite? Round the distance to the nearest tenth of a mile and the direction to the nearest degree. PRACTICE AND PROBLEM SOLVING Independent Practice Write each vector in component form. For See Exercises Example 18. JK with J (-6, -7) and K (3, -5)  18–20 21–23 24–26 27–30 31 1 2 3 4 5 TEKS TEKS TAKS TAKS Skills Practice p. S19 Application Practice p. S35 19.  EF with E (1.5, -3) and F (-2, 2.5) 20.  w Draw each vector on a coordinate plane. Find its magnitude to the nearest tenth. 21. 〈-2, 0〉 22. 〈1.5, 1.5〉 23. 〈2.5, -3.5〉 Draw each vector on a coordinate plane. Find the direction of the vector to the nearest degree. 24. A boat’s velocity is given by the vector 〈4, 1.5〉. 25. The path of a submarine is given by the vector 〈3.5, 2.5〉. 26. The path of a projectile is given by the vector 〈2, 5〉. Identify each of the following. 27. equal vectors in diagram 1 28. parallel vectors in diagram 1 29. equal vectors in diagram 2 30. parallel vectors in diagram 2 31. Aviation The pilot of a single-engine airplane flies at a constant speed of 200 km/h at a bearing of N 25° E. There is a 40 km/h crosswind blowing southeast (S 45° E). What are the

plane’s actual speed and direction? Round the speed to the nearest tenth and the direction to the nearest degree. Find each vector sum. 32. 〈1, 2〉 + 〈0, 6〉 34. 〈0, 1〉 + 〈7, 0〉 33. 〈-3, 4〉 + 〈5, -2〉 35. 〈8, 3〉 + 〈-2, -1〉 36. Critical Thinking Is vector addition commutative? That is, is  u +  v equal to  v +  u? Use the head-to-tail method of vector addition to explain why or why not. 564 564 Chapter 8 Right Triangles and Trigonometry 40°2 mi3 miCampsiteNSWEge07se_c08L06002ahiking mapGeometry 2007 SEHolt Rinehart WinstonKaren Minot(415)883-6560�� 37. This problem will prepare you for the Multi-Step TAKS Prep on page 568. A helicopter at H must fly at 50 mi/h in the direction N 45° E to reach the site of a flood victim F. There is a 41 mi/h wind in the direction N 53° W.  HX he should use so that The pilot needs to the know the velocity vector  HF. his resultant vector will be a. What is m∠F? (Hint: Consider a vertical line through F.) b. Use the Law of Cosines to find the magnitude of  HX to the nearest tenth. c. Use the Law of Sines to find m∠FHX to the nearest degree. d. What is the direction of  HX? Write each vector in component form. Round values to the nearest tenth. 38. magnitude 15, direction 42° 39. magnitude 7.2, direction 9° 40. magnitude 12.1, direction N 57° E 41. magnitude 5.8, direction N 22° E 42. Physics A classroom has a window near the ceiling,

and a long pole must be used to close it. a. Carla holds the pole at a 45° angle to the floor and applies 10 lb of force to the upper edge of the window. Find the vertical component of the vector representing the force on the window. Round to the nearest tenth. b. Taneka also applies 10 lb of force to close the window, but she holds the pole at a 75° angle to the floor. Find the vertical component of the force vector in this case. Round to the nearest tenth. c. Who will have an easier time closing the window, Carla or Taneka? (Hint: Who applies more vertical force?) 43. Probability The numbers 1, 2, 3, and 4 are written on slips of paper and placed in a hat. Two different slips of paper are chosen at random to be the x- and y-components of a vector. a. What is the probability that the vector will be equal to 〈1, 2〉? b. What is the probability that the vector will be parallel to 〈1, 2〉? 44. Estimation Use the vector 〈4, 6〉 to complete the following. a. Draw the vector on a sheet of graph paper. b. Estimate the vector’s direction to the nearest degree. c. Use a protractor to measure the angle the vector makes with a horizontal line. d. Use the vector’s components to calculate its direction. e. How did your estimate in part b compare to your measurement in part c and your calculation in part d? Multi-Step Find the magnitude of each vector to the nearest tenth and the direction of each vector to the nearest degree. 45.  u 47.  w 46.  v 48.  z 8-6 Vectors 565 565 �� 49. Football Write two vectors in component form to represent the pass pattern that Jason is told to run. Find the resultant vector and show that Jason’s move is equivalent to the vector. For each given vector, find another vector that has the same magnitude but a different direction. Then find a vector that has the same direction but a different magnitude. 50. 〈-3, 6〉 51. 〈12, 5〉 52. 〈8, -11〉 Multi-Step Find the sum of each pair

of vectors. Then find the magnitude and direction of the resultant vector. Round the magnitude to the nearest tenth and the direction to the nearest degree. 53.  u = 〈1, 2〉,  v = 〈4.8, -3.1〉  v = 〈2.5, -1〉  u = 〈-2, 7〉, 54. 55.  u = 〈6, 0〉,  v = 〈-2, 4〉 56.  u = 〈-1.2, 8〉,  v = 〈5.2, -2.1〉 57. Math History In 1827, the mathematician August Ferdinand Möbius published a book in which he introduced directed line segments (what we now call vectors). He showed how to perform scalar multiplication of vectors. For example, consider a hiker who walks along a path given by the vector walks twice as far in the same direction is given by the vector 2 a. Write the component form of the vectors b. Find the magnitude of c. Find the direction of d. Given the component form of a vector, explain how to find  v. How do they compare?  v. How do they compare?  v. The path of another hiker who  v and 2  v and 2  v and 2  v.  v. the components of the vector k  v, where k is a constant. e. Use scalar multiplication with k = -1 to write the negation of a vector  v in component form. 58. Critical Thinking A vector Another vector possible directions and magnitudes for the resultant vector.  u points due west with a magnitude of u units.  v points due east with a magnitude of v units. Describe three Math History August Ferdinand Möbius is best known for experimenting with the Möbius strip, a three-dimensional figure that has only one side and one edge. 59. Write About It Compare a line segment, a ray, and a vector. 566 566 Chapter 8 Right Triangles and Trigonometry �� 60. Which vector is parallel to 〈2, 1〉? �

� u  v  w  z 61. The vector 〈7, 9〉 represents the velocity of a helicopter. What is the direction of this vector to the nearest degree? 38° 52° 128° 142° 62. A canoe sets out on a course given by the vector 〈5, 11〉. What is the length of the canoe’s course to the nearest unit? 6 8 12 16 63. Gridded Response  AB has an initial point of (-3, 6) and a terminal point of (-5, -2). Find the magnitude of  AB to the nearest tenth. CHALLENGE AND EXTEND Recall that the angle of a vector’s direction is measured counterclockwise from the positive x-axis. Find the direction of each vector to the nearest degree. 64. 〈-2, 3〉 65. 〈-4, 0〉 66. 〈-5, -3〉 67. Navigation The captain of a ship is planning to sail in an area where there is a 4 mi/h current moving due east. What speed and bearing should the captain maintain so that the ship’s actual course (taking the current into account) is 10 mi/h at a bearing of N 70° E? Round the speed to the nearest tenth and the direction to the nearest degree. 68. Aaron hikes from his home to a park by walking 3 km at a bearing of N 30° E, then 6 km due east, and then 4 km at a bearing of N 50° E. What are the magnitude and direction of the vector that represents the straight path from Aaron’s home to the park? Round the magnitude to the nearest tenth and the direction to the nearest degree. SPIRAL REVIEW Solve each system of equations by graphing. (Previous course) x - y = -5 ⎧ 69. ⎨ ⎩ y = 3x + 1 x - 2y = 0 ⎧ 70. ⎨ ⎩ 2y + x = 8 x + y = 5 ⎧ 71. ⎨ ⎩ 3y + 15 = 2x Given that △JLM ∼ △NPS, the perimeter of △JLM is 12 cm, and the area of �

�JLM is 6 cm 2, find each measure. (Lesson 7-5) 72. the perimeter of △NPS 73. the area of △NPS Find each measure. Round lengths to the nearest tenth and angle measures to the nearest degree. (Lesson 8-5) 74. BC 75. m∠B 76. m∠C 8-6 Vectors 567 567 �� SECTION 8B Applying Trigonometric Ratios Help Is on the Way! Rescue helicopters were first used in the 1950s during the Korean War. The helicopters made it possible to airlift wounded soldiers to medical stations. Today, helicopters are used to rescue injured hikers, flood victims, and people who are stranded at sea. 1. The pilot of a helicopter is searching for an injured hiker. While flying at an altitude of 1500 ft, the pilot sees smoke at an angle of depression of 14°. Assuming that the smoke is a distress signal from the hiker, what is the helicopter’s horizontal distance to the hiker? Round to the nearest foot. 2. The pilot plans to fly due north at 100 mi/h from the helicopter’s current position H to the location of the smoke S. However there is a 30 mi/h wind in the direction N 57° W. The pilot needs to know  HA that he should use so that the velocity vector his resultant vector will be then use the Law of Cosines to find the magnitude of  HA to the nearest mile per hour.  HS. Find m∠S and 3. Use the Law of Sines to find the direction of  HA to the nearest degree. 568 568 Chapter 8 Right Triangles and Trigonometry �� SECTION 8B Quiz for Lessons 8-4 Through 8-6 8-4 Angles of Elevation and Depression 1. An observer in a blimp sights a football stadium at an angle of depression of 34°. The blimp’s altitude is 1600 ft. What is the horizontal distance from the blimp to the stadium? Round to the nearest foot. 2. When the angle of elevation of the sun is 78°, a building casts a shadow that is

6 m long. What is the height of the building to the nearest tenth of a meter? 8-5 Law of Sines and Law of Cosines Find each measure. Round lengths to the nearest tenth and angle measures to the nearest degree. 3. m∠A 4. GH 5. XZ 6. UV 7. m∠F 8. QS 8-6 Vectors Draw each vector on a coordinate plane. Find its magnitude to the nearest tenth. 9. 〈3, 1〉 10. 〈-2, -4〉 11. 〈0, 5〉 Draw each vector on a coordinate plane. Find the direction of the vector to the nearest degree. 12. A wind velocity is given by the vector 〈2, 1〉. 13. The current of a river is given by the vector 〈5, 3〉. 14. The force of a spring is given by the vector 〈4, 4〉. 15. To reach an island, a ship leaves port and sails for 6 km at a bearing of N 32° E. It then sails due east for 8 km. What are the magnitude and direction of the voyage directly from the port to the island? Round the distance to the nearest tenth of a kilometer and the direction to the nearest degree. Ready to Go On? 569 569 �� EXTENSION Trigonometry and EXTENSION the Unit Circle TEKS G.11.A Similarity and the geometry of shape: use and extend... transformations to expore... geometric figures. Also G.5.B, G.5.C, G.7.A, G.10.A, G.11.C Objective Define trigonometric ratios for angle measures greater than or equal to 90°. Rotations are used to extend the concept of trigonometric ratios to angle measures greater than or equal to 90°. Consider a ray with its endpoint at the origin, pointing in the direction of the positive x-axis. Rotate the ray counterclockwise around the origin. The acute angle formed by the ray and the nearest part of the positive or negative x-axis is called the reference angle. The rotated ray is called the terminal side of that angle. Vocabulary reference angle unit circle Angle measure: 135° Reference angle: 45° Angle measure:

345° Reference angle: 15° Angle measure: 435° Reference angle: 75° E X A M P L E 1 Finding Reference Angles Sketch each angle on the coordinate plane. Find the measure of its reference angle. A 102° B 236° Reference angle: 180° - 102° = 78° Reference angle: 236° - 180° = 56° Sketch each angle on the coordinate plane. Find the measure of its reference angle. 1a. 309° 1b. 410° The unit circle is a circle with a radius of 1 unit, centered at the origin. It can be used to find the trigonometric ratios of an angle. Consider the acute angle θ. Let P (x, y) be the point where the terminal side of θ intersects the unit circle. Draw a vertical line from P to the x-axis. Since cos θ = x __ 1 y __ 1, the coordinates of P can be written and sin θ = as (cos θ, sin θ). Thus if you know the coordinates of a point on the unit circle, you can find the trigonometric ratios for the associated angle. In trigonometry, the Greek letter theta, θ, is often used to represent angle measures. 570 570 Chapter 8 Right Triangles and Trigonometry �� E X A M P L E 2 Finding Trigonometric Ratios Find each trigonometric ratio. A cos 150° Sketch the angle on the coordinate plane. The reference angle is 30°. cos 30° = √  3 _ 2 sin 30° = 1 _ 2 Be sure to use the correct sign when assigning coordinates to a point on the unit circle. Let P (x, y) be the point where the terminal side of the angle intersects the unit circle. Since P is in Quadrant II, its x-coordinate is negative, and its y-coordinate is positive. So the coordinates of P are (- √  3 ___ 2, 1 __ 2 ). The cosine of 150° is the x-coordinate of P, so cos 150° = - √  3 ___ 2. B tan 315° Sketch the angle on the coordinate plane. The reference angle is 45°. cos 45° = √  2 _ 2 sin 45° = √  2 _ 2 Since P (x,

y) is in Quadrant IV, its y-coordinate is negative. So the coordinates of P are ( √  2 ___ 2, - √  2 ___ 2 ). Remember that tan θ = sin θ _. So tan 315° = sin 315° _ = cos θ cos 315° - √  2 ___ 2 _ √  2 ___ 2 = -1. Find each trigonometric ratio. 2a. cos 240° 2b. sin 135° EXTENSION Exercises Exercises Sketch each angle on the coordinate plane. Find the measure of its reference angle. 1. 125° 2. 216° 3. 359° Find each trigonometric ratio. 4. cos 225° 7. tan 135° 10. sin 90° 5. sin 120° 8. cos 420° 11. cos 180° 6. cos 300° 9. tan 315° 12. sin 270° 13. Critical Thinking Given that cos θ = 0.5, what are the possible values for θ between 0° and 360°? 14. Write About It Explain how you can use the unit circle to find tan 180°. 15. Challenge If sin θ ≈ -0.891, what are two values of θ between 0° and 360°? Chapter 8 Extension 571 571 �� For a complete list of the postulates and theorems in this chapter, see p. S82. Vocabulary angle of depression......... 544 equal vectors............... 561 sine........................ 525 angle of elevation........... 544 geometric mean............ 519 tangent.................... 525 component form........... 559 magnitude................. 560 trigonometric ratio......... 525 cosine..................... 525 parallel vectors......

....... 561 vector...................... 559 direction................... 560 resultant vector............. 561 Complete the sentences below with vocabulary words from the list above. 1. The? of a vector gives the horizontal and vertical change from the initial point ̶̶̶̶ to the terminal point. 2. Two vectors with the same magnitude and direction are called 3. If a and b are positive numbers, then √  ab is the?. ̶̶̶̶? of a and b. ̶̶̶̶ 4. A(n)? is the angle formed by a horizontal line and a line of sight to a point ̶̶̶̶ above the horizontal line. 5. The sine, cosine, and tangent are all examples of a(n)?. ̶̶̶̶ 8-1 Similarity in Right Triangles (pp. 518–523) E X A M P L E S EXERCISES ■ Find the geometric mean of 5 and 30. 6. Write a similarity Let x be the geometric mean. x 2 = (5)(30) = 150 Def. of geometric mean x = √150 = 5 √6 Find the positive square root. statement comparing the three triangles. TEKS G.5.B, G.5.D, G.8.A, G.8.C, G.11.A, G.11.B, G.11.C Find the geometric mean of each pair of numbers. If necessary, give the answer in simplest radical form. 7. 1 _ 4 8. 3 and 17 and 100 √  33 is the geometric mean of 3 and 3 + x. Find x, y, and z. 9. 10. ■ Find x, y, and z. ( √  33 ) 2 = 3 (3 + x) 33 = 9 + 3x 24 = 3x x = 8 y 2 = (3) (8) y 2 = 24 y = √  24 = 2 √  6 y is the geometric mean of 3 and 8. 11. z 2 = (

8) (11) z 2 = 88 z = √  88 = 2 √  22 z is the geometric mean of 8 and 11. 572 572 Chapter 8 Right Triangles and Trigonometry �� 8-2 Trigonometric Ratios (pp. 525–532) TEKS G.5.B, G.5.D, G.8.C, G.11.B, G.11.C E X A M P L E S EXERCISES Find each length. Round to the nearest hundredth. Find each length. Round to the nearest hundredth. 12. UV ■ EF sin 75° = EF _ 8.1 EF = 8.1 (sin 75°) EF ≈ 7.82 cm ■ AB tan 34° = 4.2 _ AB AB tan 34° = 4.2 AB = 4.2 _ tan 34° AB ≈ 6.23 in. Since the opp. leg and hyp. are involved, use a sine ratio. 13. PR 14. XY 15. JL Since the opp. and adj. legs are involved, use a tangent ratio. 8-3 Solving Right Triangles (pp. 534–541) TEKS G.5.D, G.7.A, G.7.C, G.8.C, G.11.B, G.11.C E X A M P L E EXERCISES ■ Find the unknown measures in △LMN. Round lengths to the nearest hundredth and angle measures to the nearest degree. Find the unknown measures. Round lengths to the nearest hundredth and angle measures to the nearest degree. 16. The acute angles of a right triangle are 17. complementary. So m∠N = 90° - 61° = 29°. sin L = MN _ Write a trig. ratio. LN sin 61° = 8.5 _ LN LN = 8.5 _ sin 61° tan L = MN _ LM tan 61° = 8.5 _ LM LM = 8.5 _ tan 61° ≈ 9.72 ≈ 4.71 Substitute the given 18. 19. values. Solve for LN. Write a trig. ratio. Substitute the given values. Solve for LM. Study Guide: Review 573 573 ��

�� 8-4 Angles of Elevation and Depression (pp. 544–549) TEKS G.5.D, G.11.C E X A M P L E S EXERCISES Classify each angle as an angle of elevation or angle of depression. ■ A pilot in a plane spots a forest fire on the ground at an angle of depression of 71°. The plane’s altitude is 3000 ft. What is the horizontal distance from the plane to the fire? Round to the nearest foot. tan 71° = 3000_ XF XF = 3000_ tan 71° XF ≈ 1033 ft ■ A diver is swimming at a depth of 63 ft below sea level. He sees a buoy floating at sea level at an angle of elevation of 47°. How far must the diver swim so that he is directly beneath the buoy? Round to the nearest foot. tan 47° = 63 _ XD XD = 63 _ tan 47° XD ≈ 59 ft 20. ∠1 21. ∠2 22. When the angle of elevation to the sun is 82°, a monument casts a shadow that is 5.1 ft long. What is the height of the monument to the nearest foot? 23. A ranger in a lookout tower spots a fire in the distance. The angle of depression to the fire is 4°, and the lookout tower is 32 m tall. What is the horizontal distance to the fire? Round to the nearest meter. 8-5 Law of Sines and Law of Cosines (pp. 551–558) TEKS G.5.B, G.5.D, G.7.A, G.11.A, G.11.C E X A M P L E S EXERCISES Find each measure. Round lengths to the nearest tenth and angle measures to the nearest degree. ■ m∠B Find each measure. Round lengths to the nearest tenth and angle measures to the nearest degree. 24. m∠Z sin B _ AC sin B _ 6 = sin C _ AB = sin 88° _ 8 Law of Sines Substitute the given values. 25. MN sin B = 6 sin 88° _ m∠B = sin -1 ( 6 sin 88° _ 8 8 ) ≈ 49° Multiply both sides by 6. 574 574 Chapter 8 Right Triangles and Trigonometry ��

�� Find each measure. Round lengths to the nearest tenth and angle measures to the nearest degree. ■ HJ Find each measure. Round lengths to the nearest tenth and angle measures to the nearest degree. 26. EF Use the Law of Cosines. 27. m∠Q HJ 2 = GH 2 + G J 2 - 2 (GH) (GJ) cos G =10 ) (11 ) cos 32° H J 2 ≈ 34.4294 HJ ≈ 5.9 Simplify. Find the square root. 8-6 Vectors (pp. 559–567) TEKS G.1.B, G.7.A, G.7.C, G.11.C E X A M P L E S EXERCISES ■ Draw the vector 〈-1, 4〉 on a coordinate plane. Find its magnitude to the nearest tenth. ⎜〈-1, 4〉⎟ = √ (-1)2 + (4)2 Write each vector in component form. 28.  AB with A (5, 1) and B (-2, 3) 29.  MN with M (-2, 4) and N (-1, -2) = √17 ≈ 4.1 30.  RS ■ The velocity of a jet is given by the vector 〈4, 3〉. Draw the vector on a coordinate plane. Find the direction of the vector to the nearest degree. In △PQR, tan P = 3 _ 4, so m∠P = tan -1 ( 3 _ ) ≈ 37°. 4 ■ Susan swims across a river at a bearing of N 75° E at a speed of 0.5 mi/h. The river’s current moves due east at 1 mi/h. Find Susan’s actual speed to the nearest tenth and her direction to the nearest degree. cos 15° = x _ 0.5 y _ 0.5 sin 15° =, so x ≈ 0.48., so y ≈ 0.13. Susan’s vector is

〈0.48, 0.13〉. The current is 〈1, 0〉. Susan’s actual speed is the magnitude of the resultant vector, 〈1.48, 0.13〉. ⎜〈1.48, 0.13〉⎟ = Her direction is tan -1 ( 0.13 _ √  (1.48) 2 + (0.13) 2 ≈ 1.5 mi/h ) ≈ 5°, or N 85° E. 1.48 Draw each vector on a coordinate plane. Find its magnitude to the nearest tenth. 31. 〈-5, -3〉 32. 〈-2, 0〉 33. 〈4, -4〉 Draw each vector on a coordinate plane. Find the direction of the vector to the nearest degree. 34. The velocity of a helicopter is given by the vector 〈4, 5〉. 35. The force applied by a tugboat is given by the vector 〈7, 2〉. 36. A plane flies at a constant speed of 600 mi/h at a bearing of N 55° E. There is a 50 mi/h crosswind blowing due east. What are the plane’s actual speed and direction? Round the speed to the nearest tenth and the direction to the nearest degree. Study Guide: Review 575 575 �� Find x, y, and z. 1. 2. 3. Use a special right triangle to write each trigonometric ratio as a fraction. 4. cos 60° 5. sin 45° 6. tan 60° Find each length. Round to the nearest hundredth. 7. PR 8. AB 9. FG 10. Nate built a skateboard ramp that covers a horizontal distance of 10 ft. The ramp rises a total of 3.5 ft. What angle does the ramp make with the ground? Round to the nearest degree. 11. An observer at the top of a skyscraper sights a tour bus at an angle of depression of 61°. The skyscraper is 910 ft tall. What is the horizontal distance from the base of the skyscraper to the tour bus? Round to the nearest foot. Find each measure

. Round lengths to the nearest tenth and angle measures to the nearest degree. 12. m∠B 13. RS 14. m∠M Draw each vector on a coordinate plane. Find its magnitude to the nearest tenth. 15. 〈1, 3〉 16. 〈-4, 1〉 17. 〈2, -3〉 Draw each vector on a coordinate plane. Find the direction of the vector to the nearest degree. 18. The velocity of a plane is given by the vector 〈3, 5〉. 19. A wind velocity is given by the vector 〈4, 1〉. 20. Kate is rowing across a river. She sets out at a bearing of N 40° E and paddles at a constant rate of 3.5 mi/h. There is a 2 mi/h current moving due east. What are Kate’s actual speed and direction? Round the speed to the nearest tenth and the direction to the nearest degree. 576 576 Chapter 8 Right Triangles and Trigonometry �� FOCUS ON SAT MATHEMATICS SUBJECT TESTS Though you can use a calculator on the SAT Mathematics Subject Tests, it may be faster to answer some questions without one. Remember to use test-taking strategies before you press buttons! 4. A swimmer jumps into a river and starts swimming directly across it at a constant velocity of 2 meters per second. The speed of the current is 7 meters per second. Given the current, what is the actual speed of the swimmer to the nearest tenth? (A) 0.3 meters per second (B) 1.7 meters per second (C) 5.0 meters per second (D) 7.3 meters per second (E) 9.0 meters per second 5. What is the approximate measure of the vertex angle of the isosceles triangle below? (A) 28.1° (B) 56.1° (C) 62.0° (D) 112.2° (E) 123.9° The SAT Mathematics Subject Tests each consist of 50 multiple-choice questions. You are not expected to have studied every topic on the SAT Mathematics Subject Tests, so some questions may be unfamiliar. You may want to time yourself as you take this practice test. It should take you

about 6 minutes to complete. 1. Let P be the acute angle formed by the line -x + 4y = 12 and the x-axis. What is the approximate measure of ∠P? (A) 14° (B) 18° (C) 72° (D) 76° (E) 85° 2. In right triangle DEF, DE = 15, EF = 36, and DF = 39. What is the cosine of ∠F? (A) 5 _ 12 (B) 12 _ 5 (C) 5 _ 13 (D) 12 _ 13 (E) 13 _ 12 3. A triangle has angle measures of 19°, 61°, and 100°. What is the approximate length of the side opposite the 100° angle if the side opposite the 61° angle is 8 centimeters long? (A) 2.5 centimeters (B) 3 centimeters (C) 9 centimeters (D) 12 centimeters (E) 13 centimeters College Entrance Exam Practice 577 577 �� Any Question Type: Estimate Once you find the answer to a test problem, take a few moments to check your answer by using estimation strategies. By doing so, you can verify that your final answer is reasonable. Gridded Response Find the geometric mean of 38 and 12 to the nearest hundredth. Let x be the geometric mean. x 2 = (38) (12) = 456 Def. of geometric mean x ≈ 21.35 Find the positive square root. Now use estimation to check that this answer is reasonable. x 2 ≈ (40) (10) = 400 Round 38 to 40 and round 12 to 10. x ≈ 20 Find the positive square root. The estimate is close to the calculated answer, so 21.35 is a reasonable answer. Multiple Choice Which of the following is equal to sin X? 0.02 0.41 0.91 2.44 Use a trigonometric ratio to find the answer. sin X = YZ _ XZ sin X = 9 _ 22 ≈ 0.41 The sine of an ∠ is opp. leg _ hyp.. Substitute the given values and simplify. Now use estimation to check that this answer is reasonable. sin X ≈ 10 _ 20 ≈ 0.5 Round 9 to 10 and round 22 to 20. The estimate is close to the calculated answer, so B is a reasonable answer. 578 578 Chapter 8 Right Triangles and Trigonometry �� An

extra minute spent checking your answers can result in a better test score. Item C Multiple Choice In △QRS, what is the ̶̶ SQ to the nearest tenth of a measure of centimeter? Read each test item and answer the questions that follow. Item A Gridded Response A cell phone tower casts a shadow that is 121 ft long when the angle of elevation to the sun is 48°. How tall is the cell phone tower? Round to the nearest foot. 1. A student estimated that the answer should be slightly greater than 121 by comparing tan 48° and tan 45°. Explain why this estimation strategy works. 2. Describe how to use the inverse tangent function to estimate whether an answer of 134 ft makes sense. Item B   BC has an initial point of Multiple Choice (-1, 0) and a terminal point of (4, 2). What are the magnitude and direction of   BC? 5.39; 22° 5.39; 68° 6.39; 22° 6.39; 68° 3. A student correctly found the magnitude of  BC as √  29. The student then calculated the value of this radical as 6.39. Explain how to use perfect squares to estimate the value of √  29. Is 6.39 a reasonable answer? 4. A student calculated the measure of the angle the vector forms with a horizontal line as 68°. Use estimation to explain why this answer is not reasonable. 9.3 centimeters 10.5 centimeters 30.1 centimeters 61.7 centimeters 5. A student calculated the answer as 30.1 cm. The student then used the diagram to estimate that SQ is more than half of RQ. So the student decided that his answer was reasonable. Is this estimation method a good way to check your answer? Why or why not? 6. Describe how to use estimation and the Pythagorean Theorem to check your answer to this problem. Item D Multiple Choice The McCleods have a variable interest rate on their mortgage. The rate is 2.625% the first year and 4% the following year. The average interest rate is the geometric mean of these two rates. To the nearest hundredth of a percent, what is the average interest rate for their mortgage? 1.38% 3.

24% 3.89% 10.50% 7. Describe how to use estimation to show that choices F and J are unreasonable. 8. To find the answer, a student uses the equation x 2 = (2.625) (4). Which compatible numbers should the student use to quickly check the answer? TAKS Tackler 579 579 �� KEYWORD: MG7 TestPrep CUMULATIVE ASSESSMENT, CHAPTERS 1–8 Multiple Choice 1. What is the length of ̶̶ UX to the nearest centimeter? 6. △ABC has vertices A (-2, -2), B (-3, 2), and C (1, 3). Which translation produces an image with vertices at the coordinates (-2, -2), (2, -1), and (-1, -6)? (x, y) → (x + 1, y - 4) (x, y) → (x + 2, y - 8) (x, y) → (x - 3, y - 5) (x, y) → (x - 4, y + 1) 7. △ABC is a right triangle in which m∠A = 30° and m∠B = 60°. Which of the following are possible lengths for the sides of this triangle? AB = √  3, AC = √  2, and BC = 1 AB = 4, AC = 2, and BC = 2 √  3 AB = 6 √  3, AC = 27, and BC = 3 √  3 AB = 8, AC = 4 √  3, and BC = 4 8. Based on the figure below, which of the following similarity statements must be true? △PQR ∼ △TSR △PQR ∼ △RTQ △PQR ∼ △TSQ △PQR ∼ △TQP 9. ABCD is a rhombus with vertices A (1, 1) and C (3, 4). Which of the following lines is parallel to diagonal ̶̶ BD? 2x - 3y = 12 2x + 3y = 12 3x + 2y = 12 3x - 4y = 12 3 centimeters 7 centimeters 9 centimeters 13 centimeters 2. △ABC is a

right triangle. m∠A = 20°, m∠B = 90°, AC = 8, and AB = 3. Which expression can be used to find BC? 3 _ tan 70° 8 _ sin 20° 8 tan 20° 3 cos 70° 3. A slide at a park is 25 ft long, and the top of the slide is 10 ft above the ground. What is the approximate measure of the angle the slide makes with the ground? 21.8° 23.6° 66.4° 68.2° 4. Which of the following vectors is equal to the vector with an initial point at (2, -1) and a terminal point at (-2, 4)? 〈-4, -5〉〈-4, 5〉〈5, -4〉〈5, 4〉 5. Which statement is true by the Addition Property of Equality? If 3x + 6 = 9y, then x + 2 = 3y. If t = 1 and s = t + 5, then s = 6. If k + 1 = ℓ + 2, then 2k + 2 = 2ℓ + 4. If a + 2 = 3b, then a + 5 = 3b + 3. 580 580 Chapter 8 Right Triangles and Trigonometry �� 10. Which of the following is NOT equivalent to sin 60°? cos 30° √  3 _ 2 (cos 60°) (tan 60°) tan 30° _ sin 30° 11. ABCDE is a convex pentagon. ∠A ≅ ∠B ≅ ∠C, ∠D ≅ ∠E, and m∠A = 2m∠D. What is the measure of ∠C? 67.5° 135° 154.2° 225° 12. Which of the following sets of lengths can represent the side lengths of an obtuse triangle? 4, 7.5, and 8.5 7, 12, and 13 9.5, 16.5, and 35 36, 75, and 88 �� Be sure to correctly identify any pairs of parallel lines before using the Alternate Interior Angles Theorem or the Same-Side Interior Angles Theorem. 13. What is the value of x? 22.5 45 90 135 Gridded Response 14. Find the next item in