Right Triangles and Trigonometry 8 • Solve problems using the geometric mean, the Pythagorean Theorem, and its converse. • Use trigonometric ratios to solve right triangle problems. • Solve triangles using the Law of Sines and the Law of Cosines. Key Vocabulary trigonometric ratio (p. 456) Law of Sines (p. 471) Law of Cosines (p. 479) Real-World Link Bridges The William H. Natcher bridge across the Ohio River has a cable-stayed design. The cables form right triangles with the supports and the length of the bridge. Right Triangles and Trigonometry Make this Foldable to help you organize your notes. Begin with seven sheets of grid paper. IN the top right corner to the bottom edge to form a square. 3 Staple the sheets along the fold in four places. 430 Chapter 8 Right Triangles and Trigonometry David Sailors/CORBIS 2 Fold the rectangular part in half. 4 Label each sheet with a lesson number and the rectangular part with the chapter title. 3IGHT5RIANGLES 1 Stack the sheets. Fold GET READY for Chapter 8 Diagnose Readiness You have two options for checking Prerequisite Skills. Option 2 Take the Online Readiness Quiz at geometryonline.com. Option 1 Take the Quick Check below. Refer to the Quick Review for help. Solve each proportion. Round to the nearest hundredth. (Lesson 7-1) 3 8 12 1. _ =_ 2. _c = _ a 4 5 f 6 d 3. _ =_ =_ 20 5 10 3 6 4 _1 4. _ =_ y= z 3 1 in. 3.5 The proportion _ =_ 5. MINIATURES x 12 in. relates the height of a miniature chair to the height of a real chair. Solve the proportion. (Lesson 7-1) EXAMPLE 1 a 31 Solve the proportion _ =_ . Round to the 5 30 nearest hundredth if necessary. 31 a _ =_ 30 Write the proportion. 5 5a = 30(31) Find the cross products. 5a = 930 Simplify. a = 186 Find the measure of the hypotenuse of each right triangle having legs with the given measures. Round to the nearest hundredth. (Extend 1-3) 6. 5 and 12 7. 6 and 8 8. 15 and 15 9. 14 and 27 10. PAINTING A ladder is propped against a wall as shown. To the nearest tenth, what is the length of the ladder? (Extend 1-3) Divide each side by 5. EXAMPLE 2 Find the measure of the hypotenuse of the right triangle having legs with the measures 10 and 24. Round to the nearest hundredth if necessary. a2 + b2 = c2 Pythagorean Theorem 102 + 242 = c2 Substitution 100 + 576 = c2 Evaluate the exponents. 676 = c2 FT Simplify. √ 676 = √c2 26 = c FT 11. The measure of one angle in a right triangle is three times the measure of the second angle. Find the measures of each angle of the triangle. Find x. (Lesson 4-2) Take the square root of each side. Simplify. EXAMPLE 3 3x˚ (x 10)˚ Find x. x˚ x + 3x + x - 10 = 180 5x = 190 x = 38 Chapter 8 Get Ready for Chapter 8 431 8-1 Geometric Mean Main Ideas • Find the geometric mean between two numbers. • Solve problems involving relationships between parts of a right triangle and the altitude to its hypotenuse. When you look at a painting, you should stand at a distance that allows you to see all of the details in the painting. The distance that creates the best view is the geometric mean of the distance from the top of the painting to eye level and the distance from the bottom of the painting to eye level. New Vocabulary geometric mean Geometric Mean The geometric mean between two numbers is the positive square root of their product. Geometric Mean You may wish to review square roots and simplifying radicals on pp. 790–791. For two positive numbers a and b, the geometric mean is the positive number x where the proportion a : x = x : b is true. This proportion can a _x 2 be written using fractions as _ x = b or with cross products as x = ab or x = √ ab . EXAMPLE Geometric Mean Find the geometric mean between each pair of numbers. b. 6 and 15 a. 4 and 9 _4 = _x Definition of geometric mean x _6 = _ Definition of geometric mean x2 = 36 Cross products x2 = 90 Cross products x 9 x 15 Take the positive square root of each side. x = √ 90 x= 6 Simplify. x = 3 √ 10 Simplify. x ≈ 9.5 1A. 5 and 45 432 Chapter 8 Right Triangles and Trigonometry Robert Brenner/PhotoEdit Take the positive square root of each side. x = √ 36 1B. 8 and 10 Use a calculator. Altitude of a −−− Triangle Consider right triangle XYZ Z with altitude WZ drawn from the right angle Z to the −− hypotenuse XY. A special relationship exists for the three right triangles, XYZ, XZW, and ZYW. X Y W GEOMETRY SOFTWARE LAB Right Triangles Formed by the Altitude Use The Geometer’s Sketchpad to draw a right triangle XYZ with right angle Z. Draw the altitude −− ZW from the right angle to the hypotenuse. Z THINK AND DISCUSS 1. Find the measures of ∠X, Y W X ∠XZY, ∠Y, ∠XWZ, ∠XZW, ∠YWZ, and ∠YZW. 2. What is the relationship between m∠X and m∠YZW? between m∠Y and m∠XZW? 3. Drag point Z to another position. Describe the relationship between the measures of ∠X and ∠YZW and between m∠Y and m∠XZW. MAKE A CONJECTURE 4. How are XYZ, XZW, and ZYW related? The Geometry Software Lab suggests the following theorem. 8.1 If the altitude is drawn from the vertex of the right angle of a right triangle to its hypotenuse, then the two triangles formed are similar to the given triangle and to each other. Altitudes of a Right Triangle The altitude drawn to the hypotenuse originates from the right angle. The other two altitudes of a right triangle are the legs. Y X Example: XYZ ∼ XWY ∼ YWZ W Z You will prove Theorem 8.1 in Exercise 38. By Theorem 8.1, since XWY ∼ YWZ, the corresponding sides are −−− −−− XW = _ YW . Notice that XW proportional. Thus, _ and ZW are segments YW ZW of the hypotenuse of the largest triangle. 8.2 The measure of an altitude drawn from the vertex of the right angle of a right triangle to its hypotenuse is the geometric mean between the measures of the two segments of the hypotenuse. Y X W Z Example: YW is the geometric mean of XW and ZW. You will prove Theorem 8.2 in Exercise 39. Lesson 8-1 Geometric Mean 433 EXAMPLE Altitude and Segments of the Hypotenuse In PQR, RS = 3 and QS = 14. Find PS. RS PS _ =_ PS QS x _3 = _ x 14 x2 = 42 P Q Theorem 8.2 S R RS = 3, QS = 14, and PS = x Cross products x ≈ 6.5 Use a calculator to take the positive square root of each side. Square Roots Since these numbers represent measures, you can ignore the negative square root value. 2. Refer to PQR above. If RS = 0.8 and QS = 2.2, find PS. ARCHITECTURE Mr. Martinez is designing a walkway to pass over a train. To find the train height, he holds a carpenter’s square at eye level and sights along the edges from the street to the top of the train. If Mr. Martinez’s eye level is 5.5 feet above the street and he is 8.75 feet from the train, find the train’s height. Round to the nearest tenth. −− Z Draw a diagram. Let YX be the altitude drawn from the right angle of WYZ. WX = _ YX _ Theorem 8.2 ZX YX 5.5 8.75 _ =_ 8.75 ZX WX = 5.5 and YX = 8.75 5.5ZX = 76.5625 ZX ≈ 13.9 Cross products Y X 5.5 ft Divide each side by 5.5. 8.75 ft W The elevated train is 5.5 + 13.9 or about 19.4 feet high. 3. Makayla is using a carpenter’s square to sight the top of a waterfall. If her eye level is 5 feet from the ground and she is a horizontal distance of 28 feet from the waterfall, find the height of the waterfall to the nearest tenth. Personal Tutor at geometryonline.com The altitude to the hypotenuse of a right triangle determines another relationship between the segments. 8.3 If the altitude is drawn from the vertex of the right angle of a right triangle to its hypotenuse, then the measure of a leg of the triangle is the geometric mean between the X measures of the hypotenuse and the segment of the hypotenuse adjacent to that leg. XZ XZ YZ XY Example: _ =_ and _ =_ XY XW YZ WZ You will prove Theorem 8.3 in Exercise 40. 434 Chapter 8 Right Triangles and Trigonometry Y W Z EXAMPLE Hypotenuse and Segment of Hypotenuse R Find x and y in PQR. −− −−− PQ and RQ are legs of right triangle PQR. Use Theorem 8.3 to write a proportion for each leg and then solve. PQ PR = _ _ PQ PS _6 = _y y 2 y2 = 12 4 S 2 RQ PR = _ _ RQ SR _6 = _x x 4 PS = 2, PQ = y, PR = 6 x2 = 24 Cross products P y Q RS = 4, RQ = x, PR = 6 Cross products y = √ 12 Take the square root. x = √ 24 Take the square root. y = 2 √ 3 Simplify. x = 2 √ 6 Simplify. y ≈ 3.5 x ≈ 4.9 Use a calculator. x Use a calculator. B 2 14 x 4. Find x and y in ABC. y A Example 1 (p. 432) Example 2 (p. 434) C Find the geometric mean between each pair of numbers. 2. 36 and 49 3. 6 and 8 4. 2 √2 and 3 √ 2 1. 9 and 4 Find the measure of the altitude drawn to the hypotenuse. 6. 5. A E 2 D 6 Example 3 (p. 434) Example 4 (p. 435) G B C H 16 F 12 7. DANCES Danielle is making a banner for the dance committee. The banner is to be as high as the wall of the gymnasium. To find the height of the wall, Danielle held a book up to her eyes so that the top and bottom of the wall were in line with the bottom edge and binding of the cover. If Danielle’s eye level is 5 feet off the ground and she is standing 12 feet from the wall, how high is the wall? Find x and y. 8. x B Extra Examples at geometryonline.com 9. C 8 D C y y 3 A B x 2 √3 D 2 A Lesson 8-1 Geometric Mean 435 HOMEWORK HELP For See Exercises Examples 10–17 1 18–23 2 24–25 3 26–31 4 Find the geometric mean between each pair of numbers. 45 and √ 80 13. √ 28 and √ 1372 10. 5 and 6 11. 24 and 25 12. √ 8 √ 3 6 √ 3 15. _ and _ 3 14. _ and 1 5 5 2 √ 2 5 √ 2 16. _ and _ 5 6 6 13 5 17. _ and _ 7 Find the measure of the altitude drawn to the hypotenuse. 19. F 20. J 18. B 5 7 8 M 16 K 12 D H 9 12 A E C 21. L G 22. V 2W Q 23. Z X 13 10 P 21 U R S 7 T 2.5 Y 25. 24. CONSTRUCTION The slope of 4 _ the roof shown below is . A 3 builder wants to put a support brace from point C perpendicular −− to AP. Find the length of the brace. N ROADS City planners want to build a road to connect points A and B. Find out how long this road will need to be. " P X 5 yd ! 4 yd MI 3 yd A C B Find x, y, and z. 26. 8 27. 28. 6 x y x z x 8 z z 3 MI y 5 y 29. 30. x y z 10 4 15 31. z y 36 6x 12 x z 8 y x The geometric mean and one extreme are given. Find the other extreme. 17 is the geometric mean between a and b. Find b if a = 7. 32. √ is the geometric mean between x and y. Find x if y = √ 33. √12 3. 436 Chapter 8 Right Triangles and Trigonometry Determine whether each statement is always, sometimes, or never true. 34. The geometric mean for consecutive positive integers is the average of the two numbers. 35. The geometric mean for two perfect squares is a positive integer. 36. The geometric mean for two positive integers is another integer. 37. The measure of the altitude of a triangle is the geometric mean between the measures of the segments of the side opposite the initial vertex. PROOF Write a proof for each theorem. 38. Theorem 8.1 39. Theorem 8.2 40. Theorem 8.3 41. RESEARCH Use the Internet or other resource to write a brief description of the golden ratio, which is also known as the divine proportion, golden mean, or golden section. EXTRA PRACTICE See pages 815, 835. Self-Check Quiz at geometryonline.com H.O.T. Problems 42. PATTERNS The spiral of the state shell of Texas, the lightning whelk, can be modeled by a geometric mean. Consider the sequence −−− −− −−− −−− −− −− −−− of segments OA, OB, OC, OD, OE, OF, OG, −−− −− −− OH, OI, and OJ. The length of each of these segments is the geometric mean between the lengths of the preceding segment and the succeeding segment. Explain this relationship. (Hint: Consider FGH.) G C J F B O H D A E I 43. OPEN ENDED Find two pairs of numbers with a geometric mean of 12. 44. REASONING Draw and label a right triangle with an altitude drawn from the right angle. From your drawing, explain the meaning of the hypotenuse and the segment of the hypotenuse adjacent to that leg in Theorem 8.3. 45. FIND THE ERROR RST is a right isosceles triangle. Holly and Ian are −− finding the measure of altitude SU. Who is correct? Explain your reasoning. Holly Ian RS _ _ = SU SU RU _ _ = SU RT 9.9 _ x _ x = 14 x2 = 138.5 138.5 x = √ x = 11.8 S SU TU _7 = _x x 7 x R x = 49 x=7 7 U 9.9 T 7 C 46. CHALLENGE Find the exact value of DE, given AD = 12 and BD = 4. A 47. D E B Writing in Math Describe how the geometric mean can be used to view paintings. Include an explanation of what happens when you are too far or too close to a painting. Lesson 8-1 Geometric Mean 437 49. REVIEW What are the solutions for the quadratic equation x2 + 9x = 36? 48. What are the values of x and y? 8 cm 6 cm y cm 10 cm F -3, -12 H 3, -12 G 3, 12 J -3, 12 x cm 50. REVIEW Tulia borrowed $300 at 15% simple interest for two years. If she makes no payments either year, how much interest will she owe at the end of the two-year period? A 4 and 6 B 2.5 and 7.5 C 3.6 and 6.4 D 3 and 7 A $90.00 C $30.00 B $45.00 D $22.50 51. The measures of the sides of a triangle are 20, 24, and 30. Find the measures of the segments formed where the bisector of the smallest angle meets the opposite side. (Lesson 7-5) A For Exercises 52 and 53, use ABC. (Lesson 7-4) G 52. If AG = 4, GB = 6, and BH = 8, find BC. B 53. If AB = 12, BC = 14, and HC = 4, find AG. H C Use the Exterior Angle Inequality Theorem to list all angles that satisfy the stated condition. (Lesson 5-2) 7 8 54. measures less than m∠8 3 1 4 2 5 55. measures greater than m∠1 56. measures less than m∠7 6 57. measures greater than m∠6 Write an equation in slope-intercept form for the line that satisfies the given conditions. (Lesson 3-4) 58. m = 2, y-intercept = 4 59. passes through (2, 6) and (-1, 0) 60. m = -4, passes through (-2, -3) 61. x-intercept is 2, y-intercept = -8 PREREQUISITE SKILL Use the Pythagorean Theorem to find the length of the hypotenuse of each right triangle. (Lesson 1-4) 62. 63. 5 ft 12 ft 64. 3 cm 5 in. 4 cm 438 Chapter 8 Right Triangles and Trigonometry 3 in. EXPLORE 8-2 Geometry Lab The Pythagorean Theorem In Chapter 1, you learned that the Pythagorean Theorem relates the measures of the legs and the hypotenuse of a right triangle. Ancient cultures used the Pythagorean Theorem before it was officially named in 1909. ACTIVITY Use paper folding to develop the Pythagorean Theorem. Step 1 On a piece of patty paper, make a mark along one side so that the two resulting segments are not congruent. Label one as a and the other as b. a Step 2 Copy these measures on the other sides in the order shown at the right. Fold the paper to divide the square into four sections. Label the area of each section. a b b ab b2 Step 3 On another sheet of patty paper, mark the same lengths a and b on the sides in the different pattern shown at the right. Step 4 Use your straightedge and pencil to connect the marks as shown at the right. Let c represent the length of each hypotenuse. b b a2 ab a b a b Step 5 Label the area of each section, which is _12ab for each triangle and c2 for the square. 1 2 ab c c 1 2 ab c2 c c 1 2 ab a Step 6 Place the squares side by side and color the corresponding regions that have the same area. For example, ab = _12ab + _12 ab. a b b a b c c 1 2 ab a c c a 1 2 ab ab b2 c2 = a2 1 2 ab ab 1 2 ab 1 2 ab The parts that are not shaded tell us that a2 + b2 = c2. ANALYZE THE RESULTS 1. Use a ruler to find actual measures for a, b, and c. Do these measures confirm that a2 + b2 = c2? 2. Repeat the activity with different a and b values. What do you notice? 3. Explain why the drawing at the right is an illustration of the Pythagorean Theorem. 4. CHALLENGE Use a geometric diagram to show that for any positive numbers a and b, a + b > √ a2 + b2. Explore 8-2 Geometry Lab: The Pythagorean Theorem 439 8-2 The Pythagorean Theorem and Its Converse Main Ideas • Use the Pythagorean Theorem. • Use the converse of the Pythagorean Theorem. New Vocabulary Pythagorean triple The Talmadge Memorial Bridge over the Savannah River, in Georgia, has two soaring towers of suspension cables. Note the right triangles being formed by the roadway, the perpendicular tower, and the suspension cables. The Pythagorean Theorem can be used to find measures in any right triangle. The Pythagorean Theorem In Lesson 1-3, you used the Pythagorean Theorem to find the distance between two points by finding the length of the hypotenuse when given the lengths of the two legs of a right triangle. You can also find the measure of any side of a right triangle given the other two measures. 8.4 Pythagorean Theorem In a right triangle, the sum of the squares of the measures of the legs equals the square of the measure of the hypotenuse. B c Symbols: a2 + b2 = c2 A b a C The geometric mean can be used to prove the Pythagorean Theorem. Proof Pythagorean Theorem Given: ABC with right angle at C Prove: a2 + b2 = C c2 a B b h y D x A c Proof: Draw right triangle ABC so C is the right angle. Then draw the altitude −− from C to AB. Let AB = c, AC = b, BC = a, AD = x, DB = y, and CD = h. 440 Chapter 8 Right Triangles and Trigonometry Alexandra Michaels/Getty Images Two geometric means now exist. _c = _a a a2 y = cy and _c = _b and b2 x b = cx Cross products Add the equations. a2 + b2 = cy + cx a2 + b2 = c(y + x) Factor. a2 + b2 = c2 Since c = y + x, substitute c for (y + x). You can use the Pythagorean Theorem to find the length of the hypotenuse or a leg of a right triangle if the other two sides are known. Find the Length of the Hypotenuse Real-World Link Due to the curvature of Earth, the distance between two points is often expressed as degree distance using latitude and longitude. This measurement closely approximates the distance on a plane. Source: NASA GEOGRAPHY California’s NASA Dryden is located at about 117 degrees longitude and 34 degrees latitude. NASA Ames, also in California, is located at about 122 degrees longitude and 37 degrees latitude. Use the lines of longitude and latitude to find the degree distance to the nearest tenth between NASA Dryden and NASA Ames. The change in longitude between the two locations is |117-122| or 5 degrees. Let this distance be a. 38˚ NASA Ames 36˚ NASA Dryden 34˚ 122˚ 120˚ 118˚ 116˚ The change in latitude is |37 - 34| or 3 degrees latitude. Let this distance be b. Use the Pythagorean Theorem to find the distance in degrees from NASA Dryden to NASA Ames, represented by c. a2 + b2 = c2 52 + 32 = c2 25 + 9 = c2 34 = c2 Pythagorean Theorem a = 5, b = 3 Simplify. Add. √ 34 = c Take the positive square root of each side. 5.8 ≈ c Use a calculator. The degree distance between NASA Dryden and NASA Ames is about 5.8 degrees. 1. GEOGRAPHY Houston, Texas, is located at about 30 degrees latitude and about 95 degrees longitude. Raleigh, North Carolina, is located at about 36 degrees latitude and about 79 degrees longitude. Find the degree distance to the nearest tenth. Personal Tutor at geometryonline.com Lesson 8-2 The Pythagorean Theorem and Its Converse StockTrek/Getty Images 441 EXAMPLE Find the Length of a Leg Find x. (XY)2 X (YZ)2 = (XZ)2 Pythagorean Theorem x2 = 142 XY = 7, XZ = 14 + 72 + 49 + x2 = 196 x2 = 147 x = √ 147 x = 7 √ 3 x ≈ 12.1 2. Find x. 14 in. 7 in. Simplify. Y x in. Z Subtract 49 from each side. Take the square root of each side. Simplify. Use a calculator. 12.1 cm x cm 16.2 cm Converse of the Pythagorean Theorem The converse of the Pythagorean Theorem can help you determine whether three measures of the sides of a triangle are those of a right triangle. 8.5 Converse of the Pythagorean Theorem If the sum of the squares of the measures of two sides of a triangle equals the square of the measure of the longest side, then the triangle is a right triangle. Symbols: If a2 + b2 = c2, then ABC is a right triangle. B c A a b C You will prove Theorem 8.5 in Exercise 30. EXAMPLE Distance Formula When using the Distance Formula, be sure to follow the order of operations carefully. Perform the operation inside the parentheses first, square each term, and then add. Verify a Triangle is a Right Triangle COORDINATE GEOMETRY Verify that PQR is a right triangle. Q (–3, 6) y R (5, 5) Use the Distance Formula to determine the lengths of the sides. P (3, 2) (-3 - 3)2 + (6 - 2)2 x1 = 3, y1 = 2, x2 = -3, y2 = 6 PQ = √ = √ (-6)2 + 42 Subtract. = √ 52 Simplify. [5 - (-3)]2 + (5 - 6)2 x1 = -3, y1 = 6, x2 = 5, y2 = 5 √ = √ 82 + (-1)2 Subtract. QR = = √ 65 PR = (5 - 3)2 + (5 - 2)2 √ Simplify. x1 = 3, y1 = 2, x2 = 5, y2 = 5 = √ 22 + 32 Subtract. = √ 13 Simplify. 442 Chapter 8 Right Triangles and Trigonometry O x By the converse of the Pythagorean Theorem, if the sum of the squares of the measures of two sides of a triangle equals the square of the measure of the longest side, then the triangle is a right triangle. PQ2 + PR2 = QR2 (√ 52 ) 2 ) - (√ 13 ) (√65 52 + 13 65 65 = 65 2 Converse of the Pythagorean Theorem 2 PQ = √ 52 , PR = √ 13 , QR = √ 65 Simplify. Add. Since the sum of the squares of two sides equals the square of the longest side, PQR is a right triangle. 3. Verify that ABC with vertices A(2, -3), B(3, 0), and C(5, -1) is a right triangle. A Pythagorean triple is three whole numbers that satisfy the equation a 2 + b 2 = c 2, where c is the greatest number. One common Pythagorean triple is 3-4-5. If the measures of the sides of any right triangle are whole numbers, the measures form a Pythagorean triple. EXAMPLE Pythagorean Triples Determine whether each set of measures can be the sides of a right triangle. Then state whether they form a Pythagorean triple. a. 8, 15, 16 Since the measure of the longest side is 16, 16 must be c, and a or b are 8 and 15, respectively. a2 + b2 = c2 82 + 152 162 64 + 225 256 289 ≠ 256 Pythagorean Theorem a = 8, b = 15, c = 16 Simplify. Add. Since 289 ≠ 256, segments with these measures cannot form a right triangle. Therefore, they do not form a Pythagorean triple. √3 √6 3 b. _, _, and _ 5 Comparing Numbers If you cannot quickly identify the greatest number, use a calculator to find decimal values for each number and compare. 5 5 a2 + b2 = c2 2 2 (_) + (_) (_) √ 3 5 √ 6 5 Pythagorean Theorem 2 √ √3 6 3 3 a=_ ,b=_ ,c=_ 5 5 5 5 3 6 9 _ +_ _ Simplify. 25 25 25 9 9 _ = _ Add. 25 25 9 9 Since _ =_ , segments with these measures form a right triangle. 25 25 However, the three numbers are not whole numbers. Therefore, they do not form a Pythagorean triple. 4A. 20, 48, and 52 Extra Examples at geometryonline.com √ √2 √ 3 5 4B. _, _, and _ 7 7 7 Lesson 8-2 The Pythagorean Theorem and Its Converse 443 Examples 1 and 2 (pp. 441–442) Find x. 1. 2. 3. 6 x 4 7 x 10 37.5 5 7 x 20 4. COMPUTERS Computer displays are usually measured along the diagonal of the screen. A 14-inch display has a diagonal that measures 14 inches. If the height of the screen is 8 inches, how wide is the screen? Example 3 (p. 442) Example 4 (p. 443) HOMEWORK HELP For See Exercises Examples 9–14 1, 2 15–18 3 19–26 4 14 in. 8 in. 5. COORDINATE GEOMETRY Determine whether JKL with vertices J(-2, 2), K(-1, 6), and L(3, 5) is a right triangle. Explain. Determine whether each set of numbers can be the measures of the sides of a right triangle. Then state whether they form a Pythagorean triple. 6. 15, 36, 39 7. √ 40 , 20, 21 8. √ 44 , 8, √ 108 Find x. 9. 10. 8 x 11. 8 8 4 28 x 14 12. 20 13. 40 14. 33 x x x 25 25 15 32 x COORDINATE GEOMETRY Determine whether QRS is a right triangle for the given vertices. Explain. 15. Q(1, 0), R(1, 6), S(9, 0) 16. Q(3, 2), R(0, 6), S(6, 6) 17. Q(-4, 6), R(2, 11), S(4, -1) 18. Q(-9, -2), R(-4, -4), S(-6, -9) Determine whether each set of numbers can be the measures of the sides of a right triangle. Then state whether they form a Pythagorean triple. 19. 8, 15, 17 20. 7, 24, 25 21. 20, 21, 31 22. 37, 12, 34 √ 74 1 ,_ 1, _ 23. _ 5 7 35 √3 √ 2 35 24. _, _, _ 2 3 36 3 _ 25. _ , 4, 1 4 5 6 _ 10 26. _ , 8, _ 7 7 7 27. GARDENING Scott wants to plant flowers in a triangular plot. He has three lengths of plastic garden edging that measure 20 inches, 21 inches, and 29 inches. Discuss whether these pieces form a right triangle. Explain. 444 Chapter 8 Right Triangles and Trigonometry Getty Images 28. NAVIGATION A fishing trawler off the coast of Alaska was ordered by the U.S. Coast Guard to change course. They were to travel 6 miles west and then sail 12 miles south to miss a large iceberg before continuing on the original course. How many miles out of the way did the trawler travel? y 29. PROOF Use the Pythagorean Theorem and the figure at the right to prove the Distance Formula. A(x1, y1) d B (x2, y2) C (x1, y2) 30. PROOF Write a paragraph proof of Theorem 8.5. x O 31. Find the value of x in the figure shown. X Real-World Career Military All branches of the military use navigation. Some of the jobs using navigation include radar/sonar operators, boat operators, airplane navigators, and space operations officers. GEOGRAPHY For Exercises 32 and 33, use the following information. Denver is located at about 105° longitude and 40° latitude. San Francisco is located at about 122° longitude and 38° latitude. Las Vegas is located at about 115° longitude and 36° latitude. Using the lines of longitude and latitude, find each degree distance. 32. San Francisco to Denver 33. Las Vegas to Denver 34. SAILING The mast of a sailboat is supported by wires called shrouds. What is the total length of wire needed to form these shrouds? 12 ft 125° 120° 115° 110° 105° 100° 40° San Francisco 35° Denver Las Vegas 30° 35. LANDSCAPING Six congruent square stones are arranged in an L-shaped walkway through a garden. If x = 15 inches, then find the area of the L-shaped walkway. 3 ft For more information, go to 26 ft shrouds x geometryonline.com. 9 ft EXTRA PRACTICE See pages 815, 835. Self-Check Quiz at geometryonline.com H.O.T. Problems 36. PAINTING A painter sets a ladder up to reach the bottom of a second-story window 16 feet above the ground. The base of the ladder is 12 feet from the house. While the painter mixes the paint, a neighbor’s dog bumps the ladder, which moves the base 2 feet farther away from the house. How far up the side of the house does the ladder reach? 37. FIND THE ERROR Maria and Colin are determining whether 5-12-13 is a Pythagorean triple. Who is correct? Explain your reasoning. Colin ? 13 + 5 2 = 12 2 ? 169 + 25 = 144 193 =/ 144 no 2 x 2 ft 12 ft Maria 5 + 122 = 132 25 + 144 = 169 169 = 169 yes 2 Lesson 8-2 The Pythagorean Theorem and Its Converse Phil Mislinski/Getty Images 16 ft 445 38. OPEN ENDED Draw a pair of similar right triangles. Are the measures of the sides of each triangle a Pythagorean triple? Explain. 39. REASONING True or false? Any two right triangles with the same hypotenuse have the same area. Explain your reasoning. 40. CHALLENGE The figure at the right is a rectangular prism with AB = 8, BC = 6, and BF = 8. Find HB. ( % & $ 41. - ! Writing in Math " About how many flags is Miko going to place? B 75 43. A rectangle has an area of 25 square inches. If the dimensions of the rectangle are doubled, what will be the area of the new rectangle? F 12.5 in 2 G 50 in 2 FT FT C 67 D 45 H 100 in 2 J 625 in 2 44. REVIEW Which equation is equivalent to 5(3 - 2x) = 7 - 2(1 - 4x)? A 18x = 10 B 2x = -10 C 2x = 10 D 10x = -10 Find the geometric mean between each pair of numbers. (Lesson 8-1) 45. 3 and 12 46. 9 and 12 47. 11 and 7 48. 6 and 9 49. GARDENS A park has a garden plot shaped like a triangle. It is bordered by a path. The triangle formed by the outside edge of the path is similar to the triangular garden. The perimeter of the outside edge of the path is 53 feet, the longest edge is 20 feet. The longest edge of the garden plot is 12 feet. What is the perimeter of the garden? (Lesson 7-5) 50. Could the sides of a triangle have the lengths 12, 13, and 25? Explain. (Lesson 5-4) PREREQUISITE SKILL Simplify each expression by rationalizing the denominator. (Pages 790–791) 7 51. _ √ 3 # Explain how right triangles are used to build suspension bridges. Which parts of the right triangle are formed by the cables? 42. Miko is going to rope off an area of the park for an upcoming concert. He is going to place a plastic flag for every three feet of rope. A 82 ' 18 52. _ √ 2 √ 14 53. _ 446 Chapter 8 Right Triangles and Trigonometry √ 2 3 √ 11 54. _ √ 3 24 55. _ √ 2 EXPLORE 8-3 Geometry Lab Patterns in Special Right Triangles Triangles with angles 45°-45°-90° measuring or 30°-60°-90° are called special right triangles. There are patterns in the measures of the sides of these triangles. ACTIVITY 1 Identify patterns in 45°-45°-90° triangles. Step 1 Draw a square with sides 4 centimeters long. Label the vertices A, B, C, and D. −− Step 2 Draw the diagonal AC. Step 3 Use a protractor to measure ∠CAB and ∠ACB. ! " $ # Step 4 Use the Pythagorean Theorem to find AC. Write in simplest form. ANALYZE THE RESULTS 1. Repeat the activity for squares with sides 6 centimeters long and 8 centimeters long. 2. MAKE A CONJECTURE What is the length of the hypotenuse of a 45°-45°-90° triangle with legs that are n units long? ACTIVITY 2 Identify patterns in 30°-60°-90° triangles. Step 1 Construct an equilateral triangle with sides 2 inches long. Label the vertices F, G, and H. −− Step 2 Find the midpoint of FH and label it J. −− Draw median GJ. ' & Step 3 Use a protractor to measure ∠FGJ, ∠F, and ∠GJF. ( * Step 4 Use the Pythagorean Theorem to find GJ. Write in simplest form. ANALYZE THE RESULTS 3. Repeat the activity to complete a table like the one at the right. 4. MAKE A CONJECTURE What are the lengths of the long leg and the hypotenuse of a 30°-60°-90° triangle with a short leg n units long? FG FJ GJ 2 in. 4 in. 5 in. Explore 8-3 Geometry Lab: Patterns in Special Right Triangles 447 8-3 Special Right Triangles Main Ideas • Use properties of 45°-45°-90° triangles. • Use properties of 30°-60°-90° triangles. Many quilt patterns use half square triangles to create a design. The pinwheel design was created with eight half square triangles rotated around the center. The measures of the angles in the half square triangles are 45°, 45°, and 90°. Properties of 45°-45°-90° Triangles Facts about 45°-45°-90° triangles are used to solve many geometry problems. The Pythagorean Theorem allows us to discover special relationships that exist among the sides of a 45°-45°-90° triangle. Draw a diagonal of a square. The two triangles formed are isosceles right triangles. Let x represent the measure of each side and let d represent the measure of the hypotenuse. d2 = x2 + x2 Pythagorean Theorem d2 = 2x2 2 d = √2x Add. x x d Take the positive square root of each side. d = √ 2 · √ x2 Factor. d = x √ 2 Simplify. This algebraic proof verifies that the length of the hypotenuse of any 2 times the length of its leg. The ratio of the sides 45°-45°-90° triangle is √ 2. is 1 : 1 : √ 8.6 In a 45°-45°-90° triangle, the length of the hypotenuse is √ 2 times the length of a leg. N N N You can use this relationship to find the measure of the hypotenuse of a 45°-45°-90° triangle given the measure of a leg of the triangle. 448 Chapter 8 Right Triangles and Trigonometry Pin Wheel Quilt. American, 19th century. Private Collection/Bridgeman Art Library EXAMPLE Find the Measure of the Hypotenuse WALLPAPER TILING Assume that the length of one of the legs of the 45°-45°-90° triangles in the wallpaper in the figure is 4 inches. What is the length of the diagonal of the entire wallpaper square? The length of each leg of the 45°-45°-90° triangle is 4 inches. The length of the hypotenuse is √ 2 times as long as a leg. So, the length of the hypotenuse of one of the triangles is 4 √2. There are four 45°-45°-90° triangles along the diagonal of the square. So, 2 ) or 16 √2 inches. the length of the diagonal of the square is 4(4 √ 1. The length of the leg of a 45°-45°-90° triangle is 7 centimeters. What is the length of the hypotenuse? EXAMPLE Find the Measure of the Legs C Find x. Rationalizing Denominators To rationalize a denominator, multiply the fraction by 1 in the form of a radical over itself so that the product in the denominator is a rational number. xm The length of the hypotenuse of a 45°-45°-90° triangle 2 times the length of a leg of the triangle. is √ xm 45˚ 45˚ B A 6m AB = (AC) √2 6 = x √ 2 6 _ =x 6 _ · √ 2 √ 2 √ 2 _ =x √ 2 √ 6 2 _ =x 2 AB = 6, AC = x Divide each side by √ 2. Rationalize the denominator. Multiply. 3 √ 2=x Divide. 4.24 ≈ x Use a calculator. 2. Refer to ABC. Suppose BA = 5m. Find x. Properties of 30°-60°-90° Triangles There is also a special relationship among the measures of the sides of a 30°-60°-90° triangle. When an altitude is drawn from any vertex of an equilateral triangle, two congruent 30°-60°-90° −−− −−− triangles are formed. LM and KM are congruent segments, so let LM = x and KM = x. By the Segment Addition Postulate, LM + KM = KL. Thus, KL = 2x. Since JKL is an equilateral triangle, KL = JL = JK. Therefore, JL = 2x and JK = 2x. Extra Examples at geometryonline.com J 30˚ 30˚ 2x L 60˚ x 2x a M Lesson 8-3 Special Right Triangles x 60˚ K 449 Let a represent the measure of the altitude. Use the Pythagorean Theorem to find a. (JM)2 + (LM)2 = (JL)2 a2 + x2 = (2x)2 a2 + x2 = 4x2 a2 = 3x2 a = √ 3x2 a = √ 3 · √ x2 a = x √ 3 Pythagorean Theorem JM = a, LM = x, JL = 2x Simplify. Subtract x2 from each side. Take the positive square root of each side. Factor. Simplify. So, in a 30°-60°-90° triangle, the measures of the sides are x, x √ 3 , and 2x. The ratio of the sides is 1: √3 : 2. The relationship of the side measures leads to Theorem 8.7. 8.7 30°-60°-90° Triangle In a 30°-60°-90° triangle, the length of the hypotenuse is twice the length of the shorter leg, and the length of the longer leg is √ 3 times the length of the shorter leg. 30˚ n√3 2n The shorter leg is opposite the 30° angle, and the longer leg is opposite the 60° angle. 60˚ EXAMPLE n 30°-60°-90° Triangles Find the missing measures. a. If BC = 14 inches, find AC. −− −− −− AC is the longer leg, AB is the shorter leg, and BC is the hypotenuse. 1 AB = _ (BC) B 60˚ 2 1 =_ (14) or 7 2 BC = 14 A AC = √ 3 (AB) = √3(7) or 7 √ 3 AB = 7 ≈ 12.12 AC is 7 √3 or about 12.12 inches. b. If AC = 8 inches, find BC. AC = √3(AB) 8 = √3(AB) 8 _ = AB √ 3 √ 8 3 _ = AB 3 BC = 2AB ( ) 8 √3 =2 _ 3 √ 16 3 =_ 3 ≈ 9.24 16 √ 3 3 BC is _ or about 9.24 inches. 3. Refer to ABC. Suppose AC = 12 in. Find BC. 450 Chapter 8 Right Triangles and Trigonometry C EXAMPLE Checking Reasonableness of Results To check the coordinates of P in Example 4, use a protractor to −− draw DP such that m∠CDP = 30. Then from the graph, you can estimate the coordinates of P. Special Triangles in a Coordinate Plane COORDINATE GEOMETRY Triangle PCD is a 30°-60°-90° triangle with right −− angle C. CD is the longer leg with endpoints C(3, 2) and D(9, 2). Locate point P in Quadrant I. −−− −− y CD lies on a horizontal gridline. Since PC −−− will be perpendicular to CD, it lies on a −−− vertical gridline. Find the length of CD. CD = |9 - 3| = 6 −−− −− CD is the longer leg. PC is the shorter leg. C (3, 2) So, CD = √3(PC). Use CD to find PC. CD = √ 3 (PC) 6 = √ 3 (PC) CD = 6 6 _ = PC 6 _ · √ 3 √3 √3 _ x O P Divide each side by √ 3. 60˚ x = PC D (9, 2) 2x Rationalize the denominator. √3 6 √3 _ = PC 3 30˚ x √3 ⫽ 6 C D Multiply. 2 √ 3 = PC Simplify. Point P has the same x-coordinate as C. P is located 2 √3 units above C. So, the coordinates of P are (3, 2 + 2 √3) or about (3, 5.46). −− 4. Triangle RST is a 30°-60°-90° triangle with right angle RST. ST is the shorter leg with endpoints S(1, 1) and T(4, 1). Locate point R in Quadrant I. Personal Tutor at geometryonline.com Example 1 (p. 449) 1. SOFTBALL Find the distance from home plate to second base if the bases are 90 feet apart. 2nd Base 90 ft 3rd Base 45˚ 45˚ d 90 ft 1st Base 45˚ 45˚ 90 ft 90 ft Home Plate Example 2 (p. 449) Find x and y. 2. 45˚ 3. y 3 x y 30˚ 8 x B Example 3 (p. 450) Find the missing measures. 4. If c = 8, find a and b. 5. If b = 18, find a and c. a 60˚ C c b 30˚ Lesson 8-3 Special Right Triangles A 451 Example 4 (p. 451) HOMEWORK HELP For See Exercises Examples 8–10, 2 18, 20 11–17, 19, 3 21–23 24–27 4 28–32 1 −− Triangle ABD is a 30°-60°-90° triangle with right angle B and with AB as the shorter leg. Graph A and B, and locate point D in Quadrant I. 6. A(8, 0), B(8, 3) 7. A(6, 6), B(2, 6) Find x and y. 8. x˚ y 9.6 x 9. y˚ 10. 5 y 17 45˚ x 11. 12. x 60˚ 13. x 18 60˚ y 60˚ 12 y 11 30˚ 60˚ y x For Exercises 14 and 15, use the figure at the right. 14. If a = 10 √3, find CE and y. 15. If x = 7 √ 3 , find a, CE, y, and b. B x E 60˚ c y a C b 16. The length of an altitude of an equilateral triangle is 12 feet. Find the length of a side of the triangle. 30˚ A 17. The perimeter of an equilateral triangle is 45 centimeters. Find the length of an altitude of the triangle. 18. The length of a diagonal of a square is 22 √2 millimeters. Find the perimeter of the square. 19. The altitude of an equilateral triangle is 7.4 meters long. Find the perimeter of the triangle. 20. The diagonals of a rectangle are 12 inches long and intersect at an angle of 60°. Find the perimeter of the rectangle. 21. The sum of the squares of the measures of the sides of a square is 256. Find the measure of a diagonal of the square. −− −− 22. Find x, y, z, and the perimeter 23. If PQ SR, find a, b, c, and d. of trapezoid ABCD. y $ P (a, b) Q (8√3, d ) # 6√3 ! X Y Z " 60˚ S ( 0, 0) T (c, 0) U (e, 0) 24. PAB is a 45°-45°-90° triangle with right angle B. Find the coordinates of P in Quadrant I for A(-3, 1) and B(4, 1). 25. PGH is a 45°-45°-90° triangle with m∠P = 90°. Find the coordinates of P in Quadrant I for G(4,-1) and H(4, 5). 452 Chapter 8 Right Triangles and Trigonometry 45˚ R (f, 0) x −−− 26. PCD is a 30°-60°-90° triangle with right angle C and CD the longer leg. Find the coordinates of P in Quadrant III for C(-3, -6) and D(-3, 7). −−− 27. PCD is a 30°-60°-90° triangle with m∠C = 30 and hypotenuse CD. −−− Find the coordinates of P for C(2, -5) and D(10, -5) if P lies above CD. Real-World Link Triangle Tiling Buildings in Federation Square in Melbourne, Australia, feature a tiling pattern called a pinwheel tiling. The sides of each right triangle are in the ratio 1 : 2 : √ 5. Source: www.federationsquare. com.au TRIANGLE TILING For Exercises 28–31, use the following information. Triangle tiling refers to the process of taking many copies of a single triangle and laying them next to each other to fill an area. For example, the pattern shown is composed of tiles like the one outlined. 28. How many 30°-60°-90° triangles are used to create the basic pattern, which resembles a circle? 29. Which angle of the 30°-60°-90° triangle is being rotated to make the basic shape? 30. Explain why there are no gaps in the basic pattern. 31. Use grid paper to cut out 30°-60°-90° triangles. Color the same pattern on each triangle. Create one basic figure that would be D part of a wallpaper tiling. 32. BASEBALL The diagram at the right shows some dimensions of U.S. Cellular Field in Chicago, −− Illinois. BD is a segment from home plate to −− dead center field, and AE is a segment from the left field foul-ball pole to the right field foul-ball pole. If the center fielder is standing at C, how far is he from home plate? 33. Find x, y, and z. EXTRA PRACTICE See pages 815, 835. x B D C B 36. In regular hexagon UVWXYZ, each side is 12 centimeters long. Find WY. X W Y V 30˚ 30˚ x H.O.T. Problems 347 ft F Self-Check Quiz at geometryonline.com 30˚ 30˚ 45˚ G A 35. Each triangle in the figure is a 30°-60°-90° triangle. Find x. 4 347 ft H 45˚ 60˚ x E y E 45˚ 34. If BD = 8 √3 and m∠DHB = 60°, find BH. z 8 C A 12 U Z 37. OPEN ENDED Draw a rectangle that has a diagonal twice as long as its width. Then write an equation to find the length of the rectangle. 38. CHALLENGE Given figure ABCD, with −− −−− AB DC, m∠B = 60°, m∠D = 45°, BC = 8, and AB = 24, find the perimeter. A B 60˚ 45˚ D C Lesson 8-3 Special Right Triangles John Gollings/Courtesy Federation Square 453 39. Writing in Math Refer to the information about quilting on page 448. Describe why quilters use the term half square triangles to describe 45°-45°-90° triangles. Explain why 45°-45°-90° triangles are used in this pattern instead of 30°-60°-90° triangles. 41. Look at the right triangle below. Which of the following could be the triangle’s dimensions? 40. A ladder is propped against a building at a 30° angle. X FT What is the length of the ladder? X A 5 ft C 10 √ 3 ft F 9 H 18 √ 2 B 10 ft D 20 ft G 9 √ 2 J 36 Determine whether each set of measures contains the sides of a right triangle. Then state whether they form a Pythagorean triple. (Lesson 7-2) 42. 3, 4, 5 43. 9, 40, 41 44. 20, 21, 31 45. 20, 48, 52 46. 7, 24, 25 47. 12, 34, 37 Find x, y, and z. (Lesson 7-1) 48. x y 49. 50. 10 12 4 z z z 8 y x 15 x y 5 Write an inequality relating each pair of angles. (Lesson 5-5) 51. m∠ALK, m∠ALN 52. m∠ALK, m∠NLO 53. m∠OLK, m∠NLO 54. m∠KLO, m∠ALN A 8.5 4.7 4 K N L 6 6 4 8.5 O 55. SCALE MODELS Taipa wants to build a scale model of the Canadian Horseshoe Falls at Niagara Falls. The height is 52 meters. If she wants the model to be 80 centimeters tall, what scale factor will she use? (Lesson 7-1) PREREQUISITE SKILL Solve each equation. (Pages 781–782) x 56. 5 = _ x 57. _ = 0.14 10 58. 0.5 = _ 7 60. _ n = 0.25 m 61. 9 = _ 24 62. _ x = 0.4 3 9 0.8 454 Chapter 8 Right Triangles and Trigonometry k 13 59. 0.2 = _ g 35 63. _ y = 0.07 6.5 EXPLORE 8-4 Graphing Calculator Lab Trigonometry You have investigated the patterns in the measures of special right triangles. The study of the patterns in all right triangles is called trigonometry. You can use the Cabri Junior application on a TI-83/84 Plus to investigate these patterns. ACTIVITY Step 1 Use the line tool on the F2 menu to draw a line. Label the points on the line A and B. Step 2 Press F3 and choose the Perpendicular tool to create a perpendicular line through point B. Draw and label a point C on the perpendicular. Step 3 Use the segment tool on the F2 menu to −− draw AC. −− −− Step 4 Find and label the measures of BC and AC using the Distance and Length tool under Measure on the F5 menu. Use the Angle tool for the measure of ∠A. BC Step 6 Calculate and display the ratio _ using the AC Calculate tool on the F5 menu. Label the ratio as A/B. Step 7 Press CLEAR . Then use the arrow keys to move the cursor close to point B. When the arrow is clear, press and hold the ALPHA key. Drag B and observe the ratio. ANALYZE THE RESULTS 1. Discuss the effect of dragging point B on BC, AC, m∠A, and the BC ratio _ . AC AB BC 2. Use the calculate tool to find the ratios _ and _ . Then drag B and AB AC observe the ratios. 3. MAKE A CONJECTURE The sine, cosine, and tangent functions are trigonometric functions based on angle measures. Make a note of m∠A. Exit Cabri Jr. and use SIN , COS and TAN on the calculator to find sine, cosine and tangent for m∠A. Compare the results to the ratios you found in the activity. Make a conjecture about the definitions of sine, cosine, and tangent. Explore 8-4 Graphing Calculator Lab: Trigonometry 455 8-4 Trigonometry Main Ideas • Find trigonometric ratios using right triangles. • Solve problems using trigonometric ratios. New Vocabulary trigonometry trigonometric ratio sine cosine tangent The branch of mathematics known as trigonometry was developed for use by astronomers and surveyors. Surveyors use an instrument called a theodolite (thee AH duh lite) to measure angles. It consists of a telescope mounted on a vertical axis and a horizontal axis. After measuring the angles, surveyors apply trigonometry to calculate distance or height. Trigonometric Ratios The word trigonometry comes from two Greek terms, trigon, meaning triangle, and metron, meaning measure. The study of trigonometry involves triangle measurement. A ratio of the lengths of sides of a right triangle is called a trigonometric ratio. The three most common trigonometric ratios are sine, cosine, and tangent. Trigonometric Ratios Words Symbols leg opposite ∠A sine of ∠A = __ BC sin A = _ leg opposite ∠B sine of ∠B = __ AC sin B = _ hypotenuse hypotenuse AB AC cos A = _ leg adjacent to ∠B cosine of ∠B = __ BC cos B = _ hypotenuse A AB BC tan A = _ leg opposite ∠B tangent of ∠B = __ leg adjacent to ∠B AC tan B = _ BC hypotenuse leg opposite ⬔A leg opposite ⬔B C B AB leg opposite ∠A tangent of ∠A = __ leg adjacent to ∠A B AB leg adjacent to ∠A cosine of ∠A = __ hypotenuse Models hypotenuse A leg adjacent to ⬔A leg adjacent to ⬔B C B leg AC hypotenuse A leg adjacent to ⬔A and opposite ⬔B opposite ⬔A and adjacent to C ⬔B Trigonometric ratios are related to the acute angles of a right triangle, not the right angle. 456 Chapter 8 Right Triangles and Trigonometry Arthur Thevenart/CORBI Reading Math Memory Hint SOH-CAHTOA is a mnemonic device for learning the ratios for sine, cosine, and tangent using the first letter of each word in the ratios. opp hyp adj cos A = _ hyp opp tan A = _ adj EXAMPLE T Find Sine, Cosine, and Tangent Ratios 3 4 Find sin R, cos R, tan R, sin S, cos S, and tan S. Express each ratio as a fraction and as a decimal. S sin R = __ opposite leg hypotenuse ST =_ RS 4 or 0.8 =_ 5 cos R = _ adjacent leg hypotenuse RT =_ RS 3 =_ or 0.6 5 tan R = __ opposite leg hypotenuse RT _ = RS 3 or 0.6 =_ 5 cos S = _ adjacent leg hypotenuse ST _ = RS 4 or 0.8 =_ 5 tan S = __ sin A = _ sin S = __ R 5 opposite leg adjacent leg ST =_ RT 4 or 1.− =_ 3 3 opposite leg adjacent leg RT _ = ST 3 = _ or 0.75 4 K 13 1. Find sin J, cos J, tan J, sin K, cos K, and tan K. Express each ratio as a fraction and as a decimal. J 5 12 L You can use paper folding to investigate trigonometric ratios in similar right triangles. GEOMETRY LAB A Trigonometric Ratios • Fold a rectangular piece of paper along a diagonal from A to C. Then cut along the fold to form right triangle ABC. Write the name of each angle on the inside of the triangle. A B B −− • Fold the triangle so that there are two segments perpendicular to BA. Label −− −− −− −− −− points D, E, F, and G as shown. Use a ruler to measure AC, AB, BC, AF, AG, −− −− −− −− FG, AD, AE, and DE to the nearest millimeter. ANALYZE THE RESULTS 1. What is true of AED, AGF, and ABC? 2. Copy the table. Write the ratio of the side lengths for each ratio. Then calculate a C C A ED G B F C value for each ratio to the nearest ten-thousandth. In AED In AGF In ABC sin A cos A tan A 3. Study the table. Write a sentence about the patterns you observe. 4. What is true about m∠A in each triangle? Extra Examples at geometryonline.com Lesson 8-4 Trigonometry 457 As the Geometry Lab suggests, the value of a trigonometric ratio depends only on the measure of the angle. It does not depend on the size of the triangle. EXAMPLE Graphing Calculator Be sure your calculator is in degree mode rather than radian mode. Your calculator may require you to input the angle before using the trigonometric key. Use a Calculator to Evaluate Expressions Use a calculator to find cos 39° to the nearest ten-thousandth. KEYSTROKES: COS 39 ENTER cos 39° ≈ 0.7771 2. sin 67° Use Trigonometric Ratios You can use trigonometric ratios to find the missing measures of a right triangle if you know the measures of two sides of a triangle or the measure of one side and one acute angle. EXAMPLE Use Trigonometric Ratios to Find a Length SURVEYING Dakota is standing on the ground 97 yards from the base of a cliff. Using a theodolite, he noted that the angle formed by the ground and the line of sight to the top of the cliff is 56°. Find the height of the cliff to the nearest yard. Let x be the height of the cliff in yards. leg opposite x tan 56° = _ tan = _ leg adjacent x yd 97 97 tan 56° = x Multiply each side by 97. 56˚ Use a calculator to find x. KEYSTROKES: 97 yd 97 TAN 56 ENTER 143.8084139 The cliff is about 144 yards high. 3. MEASUREMENT Jonathan is standing 15 yards from a roller coaster. The angle formed by the ground to the top of the roller coaster is 71°. How tall is the roller coaster? When solving equations like 3x = -27, you use the inverse of multiplication to find x. In trigonometry, you can find the measure of the angle by using the inverse of sine, cosine, or tangent. Given equation To find the angle sin A = x A= sin-1 (x) A equals the inverse sine of x. A= cos-1 (y) A equals the inverse cosine of y. A= tan-1 (z) A equals the inverse tangent of z. cos A = y tan A = z 458 Chapter 8 Right Triangles and Trigonometry Read as EXAMPLE Calculators The second functions of the 3). , #/3 , and 4!. keys are usually the inverses. Use Trigonometric Ratios to Find an Angle Measure COORDINATE GEOMETRY Find m∠A in right triangle ABC for A(1, 2), B(6, 2), and C(5, 4). y Explore You know the coordinates of the vertices of a right triangle and that ∠C is the right angle. You need to find the measure of one of the angles. Plan Use the Distance Formula to find the measure of each side. Then use one of the trigonometric ratios to write an equation. Use the inverse to find m∠A. Solve AB = (6 - 1)2 + (2 - 2)2 √ BC = A(1, 2) O B(6, 2) x (5 - 6)2 + (4 - 2)2 √ = √ 1 + 4 or √ 5 = √ 25 + 0 or 5 AC = C(5, 4) (5 - 1)2 + (4 - 2)2 √ = √ 16 + 4 or 2 √ = √20 5 Use the cosine ratio. You can use trigonometry to help you come closer to locating the hidden treasure. Visit geometryonline.com. AC cos A = _ leg adjacent hypotenuse cos = _ AB 2 √ 5 5 cos A = _ AC = 2 √ 5 and AB = 5 ( ) 2 √ 5 A = cos-1 _ Solve for A. 5 Use a calculator to find m∠A. KEYSTROKES: ND [COS-1] 2 ND [ √ ] 5 5 %.4%2 m∠A ≈ 26.56505118 The measure of ∠A is about 26.6. Check Use the sine ratio to check the answer. BC sin A = _ sin = _ AB leg opposite hypotenuse √ 5 5 sin A = _ BC = √5 and AB = 5 KEYSTROKES: ND [SIN-1] ND [ √ ] 5 5 %.4%2 m∠A ≈ 26.56505118 The answer is correct. 4. Find m∠P in right PQR for P(2, -1), Q(4, 3), and R(8, 1). Personal Tutor at geometryonline.com Lesson 8-4 Trigonometry 459 Example 1 (p. 457) Example 2 (p. 458) Example 3 (p. 458) Use ABC to find sin A, cos A, tan A, sin B, cos B, and tan B. Express each ratio as a fraction and as a decimal to the nearest hundredth. 1. a = 14, b = 48, and c = 50 2. a = 8, b = 15, and c = 17 B c a b C A Use a calculator to find each value. Round to the nearest ten-thousandth. 4. cos 60° 5. cos 33° 3. sin 57° 6. tan 30° 7. tan 45° 8. sin 85° 9. SURVEYING Maureen is standing on horizontal ground level with the base of the CN Tower in Toronto, Ontario. The angle formed by the ground and the line segment from her position to the top of the tower is 31.2°. She knows that the height of the tower to the top of the antennae is about 1815 feet. Find her distance from the CN Tower to the nearest foot. Find the measure of each angle to the nearest tenth of a degree. 11. sin B = 0.6307 10. tan A = 1.4176 Example 4 (p. 459) HOMEWORK HELP For See Exercises Examples 14–17 1 18–23 2 24, 25 3 26–28 4 COORDINATE GEOMETRY Find the measure of the angle to the nearest tenth in each right triangle ABC. 12. ∠A in ABC, for A(6, 0), B(-4, 2), and C(0, 6) 13. ∠B in ABC, for A(3, -3), B(7, 5), and C(7, -3) P Use PQR with right angle R to find sin P, cos P, tan P, 1 sin Q, cos Q, and tan Q. Express each ratio as a fraction, R and as a decimal to the nearest hundredth. 14. p = 12, q = 35, and r = 37 15. p = √6, q = 2 √3, and r = 3 √2 3 √3 2 3 16. p = _ , q = _, and r = 3 2 17. p = 2 √3, q = √ 15 , and r = 3 √3 Use a calculator to find each value. Round to the nearest ten-thousandth. 19. tan 42.8° 20. cos 77° 18. sin 6° 21. sin 85.9° 22. tan 12.7° 23. cos 22.5° 24. AVIATION A plane is one mile above sea level when it begins to climb at a constant angle of 3° for the next 60 ground miles. About how far above sea level is the plane after its climb? 460 Chapter 8 Right Triangles and Trigonometry David R. Frazier/Photo Researchers 3˚ 60 mi 2 Q 0 25. MONUMENTS At 351 feet tall, the Jefferson Davis Monument in Fairview, Kentucky, is the largest concrete obelisk in the world. Pedro is looking at the top of the monument at an angle of 75°. How far away from the monument is he standing? COORDINATE GEOMETRY Find the measure of each angle to the nearest tenth in each right triangle. 26. ∠J in JCL for J(2, 2), C(2, -2), and L(7, -2) 27. ∠C in BCD for B(-1, -5), C(-6, -5), and D(-1, 2) 28. ∠X in XYZ for X(-5, 0), Y(7, 0), and Z(0, √ 35 ) Real-World Link The Jefferson Davis Monument in Fairview, Kentucky, is the fourth tallest monument in the United States. The walls are seven feet thick at the base, tapering to two feet thick at the top. Source: parks.ky.gov Use the figure to find each trigonometric ratio. Express answers as a fraction and as a decimal rounded to the nearest A ten-thousandth. 29. sin A 32. sin x° 35. cos B C 5√26 x˚ 25 30. tan B 33. cos x° 36. sin y° y˚ 5 √26 D B 1 31. cos A 34. tan A 37. tan x° Find the measure of each angle to the nearest tenth of a degree. 39. cos C = 0.2493 40. tan E = 9.4618 38. sin B = 0.7245 41. sin A = 0.4567 42. cos D = 0.1212 43. tan F = 0.4279 Find x. Round to the nearest tenth. 45. 44. x 46. 17 24˚ x 62˚ 12 60 19 x˚ 47. 48. 34 x 17˚ 49. 6.6 x 18 31˚ 15 x˚ SAFETY For Exercises 50 and 51, use the following information. To guard against a fall, a ladder should make an angle of 75° or less with the ground. 50. What is the maximum height that a 20-foot ladder can reach safely? 51. How far from the building is the base of the ladder at the maximum height? EXTRA PRACTICE Find x and y. Round to the nearest tenth. 53. 52. x˚ x See pages 816, 835. 24 36 Self-Check Quiz at geometryonline.com y˚ 55˚ 12 75˚ 54. C y x 24 47˚ A 32˚ y D Lesson 8-4 Trigonometry Gibson Stock Photography B 461 H.O.T. Problems 55. OPEN ENDED Draw a right triangle and label the measure of one acute angle and the measure of the side opposite that angle. Then solve for the remaining measures. 56. CHALLENGE Use the figure at the right to find sin x°. A 10 57. REASONING Explain the difference between x x -1 _ tan A = _ y and tan y = A. () 58. x˚ D Writing in Math B 8 10 Refer to the information on theodolites on page 456. Explain how surveyors determine angle measures. Include the kind of information one obtains from a theodolite. 20 59. In the figure, if cos x = _ , what are 29 sin x and tan x? 8 C 60. REVIEW What is the solution set of the quadratic equation x2 + 4x - 2 = 0? F {-2, 2} G {-2 + √ 6 , -2 - √ 6} X H {-2 + √ 2 , -2 - √ 2} J no real solution 29 29 A sin x = _ and tan x = _ 21 21 20 21 and tan x = _ B sin x = _ 29 21 61. REVIEW Which of the following has the same value as 9-15 × 93? 29 21 and tan x = _ C sin x = _ 20 21 21 and tan x = _ 21 D sin x = _ 29 A 9-45 C 9-12 B 9-18 D 9-5 20 Find each measure. (Lesson 8-3) B 62. If a = 4, find b and c. a 63. If b = 3, find a and c. C 60˚ c b 30˚ A Determine whether each set of measures can be the sides of a right triangle. Then state whether they form a Pythagorean triple. (Lesson 8-2) 64. 4, 5, 6 65. 5, 12, 13 66. 9, 12, 15 67. 8, 12, 16 68. TELEVISION During a 30-minute television program, the ratio of minutes of commercials to minutes of the actual show is 4 : 11. How many minutes are spent on commercials? (Lesson 7-1) s t PREREQUISITE SKILL Find each angle measure if h k. (Lesson 3-2) 1 69. m∠15 70. m∠7 71. m∠3 72. m∠12 73. m∠11 74. m∠4 462 Chapter 8 Right Triangles and Trigonometry 30˚ 4 7 6 5 3 2 8 9 10 11 12 117˚ 15 14 h k APTER CH 8 Mid-Chapter Quiz Lessons 8-1 through 8-4 Find the measure of the altitude drawn to the hypotenuse. (Lesson 8-1) 1. 2. X B 5 9 A 21 7 C 6. WOODWORKING Ginger made a small square table for her workshop with a diagonal that measures 55 inches. What are the measures of the sides? Recall that a square has right angles at the corners and congruent sides. (Lesson 8-3) Y Z 3. Determine whether ABC with vertices A(2, 1), B(4, 0), and C(5, 7) is a right triangle. Explain. (Lesson 8-2) 4. MULTIPLE CHOICE To get from your campsite to a trail head, you must take the path shown below to avoid walking through a pond. M Find x and y. (Lesson 8-3) 7. 8. y 45˚ x 30˚ x 9. MULTIPLE CHOICE In the right triangle, what is AB if BC = 6? (Lesson 8-3) A TRAILHEAD M y 6 3 4x˚ 2x˚ CAMPSITE C About how many meters would be saved if it were possible to walk through the pond? (Lesson 8-2) A 55.0 C 24.7 B 39.2 D 15.8 5. DOG WALKING A man is walking his dog on level ground in a straight line with the dog’s favorite tree. The angle from the man’s present position to the top of a nearby telephone pole is 45º. The angle from the tree to the top of the telephone pole is 60º. If the telephone pole is 50 feet tall, about how far is the man with the dog from the tree? (Lesson 8-3) FT B F 12 units H 4 √ 3 units G 6 √ 2 units J 2 √ 3 units Find x to the nearest tenth. (Lesson 8-4) 10. 11. x 53 16 ˚ 32 12. 13. 2 x˚ 10 x˚ x 5 9.7 17˚ 14. GARDENING The lengths of the sides of a triangular garden are 32 feet, 24 feet, and 40 feet. What are the measures of the angles formed on each side of the garden? (Lesson 8-4) Find the measure of each angle to the nearest tenth of a degree. (Lesson 8-4) 15. sin T = 0.5299 16. cos W = 0.0175 Chapter 8 Mid-Chapter Quiz 463 8-5 Angles of Elevation and Depression Main Ideas • Solve problems involving angles of elevation. • Solve problems involving angles of depression. A pilot is getting ready to take off from Mountain Valley airport. She looks up at the peak of a mountain immediately in front of her. The pilot must estimate the speed needed and the angle formed by a line along the runway and a line from the plane to the peak of the mountain to clear the mountain. New Vocabulary Angles of Elevation An angle of angle of elevation angle of depression elevation is the angle between the line of sight and the horizontal when an observer looks upward. A • B line of sight D angle of elevation EXAMPLE C Angle of Elevation AVIATION The peak of Goose Bay Mountain is 400 meters higher than the end of a local airstrip. The peak rises above a point 2025 meters from the end of the airstrip. A plane takes off from the end of the runway in the direction of the mountain at an angle that is kept constant until the peak has been cleared. If the pilot wants to clear the mountain by 50 meters, what should the angle of elevation be for the takeoff to the nearest tenth of a degree? Make a drawing. 50 m D B 400 m 2025 m A C Since CB is 400 meters and BD is 50 meters, CD is 450 meters. Let x represent m∠DAC. opposite CD tan x° = _ tan = _ AC 450 tan x° = _ 2025 adjacent CD = 450, AC = 2025 ( 2025 ) 450 x = tan-1 _ Solve for x. x ≈ 12.5 Use a calculator. The angle of elevation for the takeoff should be more than 12.5°. 1. SHADOWS Find the angle of elevation of the Sun when a 7.6-meter flagpole casts a 18.2-meter shadow. Round to the nearest tenth of a degree. 464 Chapter 8 Right Triangles and Trigonometry Extra Examples at geometryonline.com Angles of Depression An angle of angle of depression B depression is the angle between the line of sight when an observer looks downward and the horizontal. A line of sight C D Angle of Depression The tailgate of a moving van is 3.5 feet above the ground. A loading ramp is attached to the rear of the van at an incline of 10°. Which is closest to the length of the ramp? A 3.6 ft C 19.8 ft B 12.2 ft D 20.2 ft 10˚ 3.5 ft sin 10° ≈ 0.17 cos 10° ≈ 0.98 tan 10° ≈ 0.18 Read the Test Item The angle of depression between the ramp and the horizontal is 10°. Use trigonometry to find the length of the ramp. Solve the Test Item 10˚ A D 3.5 ft C 10˚ B The ground and the horizontal level with the back of the van are parallel. Therefore, m∠DAB = m∠ABC since they are alternate interior angles. opposite 3.5 sin 10° = _ sin = _ AB AB sin 10° = 3.5 3.5 AB = _ Check Results Before moving on to the next question, check the reasonableness of your answer. Analyze your result to determine that it makes sense. sin 10° 3.5 AB ≈ _ 0.17 AB ≈ 20.2 hypotenuse Multiply each side by AB. Divide each side by sin 10°. sin 10° ≈ 0.17 Divide. The ramp is about 20.2 feet long. So the correct answer is choice D. 2. HIKING Ayana is hiking in a national park. A forest ranger is standing in a fire tower that overlooks a meadow. She sees Ayana at an angle of depression measuring 38°. If Ayana is 50 feet away from the base of the tower, which is closest to the height of the fire tower? F 30.8 ft H 39.4 ft G 39.1 ft J 63.5 ft FT sin 38° ≈ 0.62 cos 38° ≈ 0.79 tan 38° ≈ 0.78 Personal Tutor at geometryonline.com Lesson 8-5 Angles of Elevation and Depression 465 Angles of elevation or depression to two different objects can be used to find the distance between those objects. EXAMPLE Common Misconception The angle of depression is often not an angle of the triangle but the complement to an angle of the triangle. In DBC, the angle of depression is ∠BCE, not ∠DCB. Indirect Measurement Olivia works in a lighthouse on a cliff. She observes two sailboats due east of the lighthouse. The angles of depression to the two boats are 33° and 57°. Find the distance between the two sailboats to the nearest foot. C 85 ft E 33˚ 57˚ 110 ft CDA and CDB are right triangles, and CD = 110 + 85 or 195. The D A distance between the boats is AB or DB - DA. Use the right triangles to find these two lengths. −− −− Because CE and DB are horizontal lines, they are parallel. Thus, ∠ECB ∠CBD and ∠ECA ∠CAD because they are alternate interior angles. This means that m∠CBD = 33 and m∠CAD = 57. B Use the measures of CBD to find DB. 195 tan 33° = _ DB DB tan 33° = 195 opposite adjacent tan = _; m∠CBD = 33 Multiply each side by DB. 195 DB = _ Divide each side by tan 33°. DB ≈ 300.27 Use a calculator. tan 33° Use the measures of CAD to find DA. 195 tan 57° = _ DA DA tan 57° = 195 opposite adjacent tan = _; m∠CAD = 57 Multiply each side by DA. 195 DA = _ tan 57° Divide each side by tan 57°. DA ≈ 126.63 Use a calculator. The distance between the boats is DB - DA. DB - DA ≈ 300.27 - 126.63 or about 174 feet 3. BOATING Two boats are observed by a parasailer 75 meters above a lake. The angles of depression are 12.5° and 7°. How far apart are the boats? Example 1 (p. 464) 1. AVIATION A pilot is flying at 10,000 feet and wants to take the plane up to 20,000 feet over the next 50 miles. What should be his angle of elevation to the nearest tenth? (Hint: There are 5280 feet in a mile.) 466 Chapter 8 Right Triangles and Trigonometry Example 2 (p. 465) Example 3 (p. 466) 2. OCEAN ARCHAEOLOGY A salvage ship uses sonar to determine the angle of depression to a wreck on the ocean floor that is 40 meters below the surface. How far must a diver, lowered from the salvage ship, walk along the ocean floor to reach the wreck? 3. STANDARDIZED TEST EXAMPLE From the top of a 150-foot high tower, an air traffic controller observes an airplane on the runway. Which equation would be used to find the distance from the base of the tower to the airplane? 150 A x = 150 tan 12° B x = _ cos 12° HOMEWORK HELP For See Exercises Examples 4–11 1 12, 13 2 14–17 3 13.25˚ FT X 150 C x=_ 150 D x=_ tan 12° 4. GOLF A golfer is standing at the tee, looking up to the green on a hill. If the tee is 36 yards lower than the green and the angle of elevation from the tee to the hole is 12°, find the distance from the tee to the hole. 5. TOURISM Crystal is on a bus in France with her family. She sees the Eiffel Tower at an angle of 27°. If the tower is 986 feet tall, how far away is the bus? Round to the nearest tenth. 40 m sin 12° 36 yd 12˚ P T 27˚ B CIVIL ENGINEERING For Exercises 6 and 7, use the following information. The percent grade of a highway is the ratio of the vertical rise or fall over a horizontal distance expressed to the nearest whole percent. Suppose a highway has a vertical rise of 140 feet for every 2000 feet of horizontal distance. 6. Calculate the percent grade of the highway. 7. Find the angle of elevation that the highway makes with the horizontal. 8. SKIING A ski run has an angle of elevation of 24.4° and a vertical drop of 1100 feet. To the nearest foot, how long is the ski run? GEYSERS For Exercises 9 and 10, use the following information. Kirk visits Yellowstone Park and Old Faithful on a perfect day. His eyes are 6 feet from the ground, and the geyser can reach heights ranging from 90 feet to 184 feet. 9. If Kirk stands 200 feet from the geyser and the eruption rises 175 feet in the air, what is the angle of elevation to the top of the spray to the nearest tenth? 10. In the afternoon, Kirk returns and observes the geyser’s spray reach a height of 123 feet when the angle of elevation is 37°. How far from the geyser is Kirk standing to the nearest tenth of a foot? Lesson 8-5 Angles of Elevation and Depression 467 11. RAILROADS Refer to the information at the left. Determine the incline of the Monongahela Incline. 12. AVIATION After flying at an altitude of 500 meters, a helicopter starts to descend when its ground distance from the landing pad is 11 kilometers. What is the angle of depression for this part of the flight? 13. SLEDDING A sledding run is 300 yards long with a vertical drop of 27.6 yards. Find the angle of depression of the run. Real-World Link The Monongahela Incline, in Pittsburgh, Pennsylvania, is 635 feet long with a vertical rise of 369.39 feet. Although opened on May 28, 1870, it is still used by commuters to and from Pittsburgh. Source: www.portauthority.org 14. AMUSEMENT PARKS From the top of a roller coaster, 60 yards above the ground, a rider looks down and sees the merry-go-round and the Ferris wheel. If the angles of depression are 11° and 8°, respectively, how far apart are the merry-go-round and the Ferris wheel? 8˚ 11˚ 60 yd 15. BIRD WATCHING Two observers are 200 feet apart, in line with a tree containing a bird’s nest. The angles of elevation to the bird’s nest are 30° and 60°. How far is each observer from the base of the tree? 16. METEOROLOGY The altitude of the base of a cloud formation is called the ceiling. To find the ceiling one night, a meteorologist directed a spotlight vertically at the clouds. Using a theodolite placed 83 meters from the spotlight and 1.5 meters above the ground, he found the angle of elevation to be 62.7°. How high was the ceiling? 62.7˚˚ 1.5 m 83 m 17. TRAVEL Kwan-Yong uses a theodolite to measure the angle of elevation from the ground to the top of Ayers Rock to be 15.85°. He walks half a kilometer closer and measures the angle of elevation to be 25.6°. How high is Ayers Rock to the nearest meter? 18. PHOTOGRAPHY A digital camera with a panoramic lens is described as having a view with an angle of elevation of 38º. If the camera is on a 3-foot tripod aimed directly at a 124-foot monument, how far from the monument should you place the tripod to see the entire monument in your photograph? MEDICINE For Exercises 19–21, use the following information. A doctor is using a treadmill to assess the strength of a patient’s heart. At the beginning of the exam, the 48-inch long treadmill is set at an incline of 10°. 19. How far off the horizontal is the raised end of the treadmill at the beginning of the exam? 20. During one stage of the exam, the end of the treadmill is 10 inches above the horizontal. What is the incline of the treadmill to the nearest degree? 21. Suppose the exam is divided into five stages and the incline of the treadmill is increased 2° for each stage. Does the end of the treadmill rise the same distance between each stage? 468 Chapter 8 Right Triangles and Trigonometry R. Krubner/H. Armstrong Roberts EXTRA PRACTICE See pages 816, 835. Self-Check Quiz at geometryonline.com H.O.T. Problems 22. AEROSPACE On July 20, 1969, Neil Armstrong became the first human to walk on the Moon. During this mission, Apollo 11 orbited the Moon three miles above the surface. At one point in the orbit, the onboard guidance system measured the angles of depression to the far and near edges of a large crater. The angles measured 16° and 29°, respectively. Find the distance across the crater. ÀLÌ £Èc Óc ÎÊ v 23. OPEN ENDED Find a real-life example of an angle of depression. Draw a diagram and identify the angle of depression. 24. REASONING Explain why an angle of elevation is given that name. 25. CHALLENGE Two weather observation stations are 7 miles apart. A weather balloon is located between the stations. From Station 1, the angle of elevation to the weather balloon is 33°. From Station 2, the angle of elevation to the balloon is 52°. Find the altitude of the balloon to the nearest tenth of a mile. Writing in Math Describe how an airline pilot would use angles of elevation and depression. Make a diagram and label the angles of elevation and depression. Then describe the difference between the two. 26. 27. The top of a signal tower is 120 meters above sea level. The angle of depression from the top of the tower to a passing ship is 25°. Which is closest to the distance from the foot of the tower to the ship? 25˚ 28. REVIEW What will happen to the slope of line p if the line is shifted so that the y-intercept decreases and the x-intercept remains the same? y 8 6 4 2 ⫺8⫺6⫺4⫺2 O 120 m 2 4 6 8x ⫺4 ⫺6 ⫺8 sin 25° ≈ 0.42 cos 25° ≈ 0.91 tan 25° ≈ 0.47 F The slope will change from negative to positive. G The slope will become undefined. A 283.9 m C 132.4 m H The slope will decrease. B 257.3 m D 56.0 m J The slope will increase. Lesson 8-5 Angles of Elevation and Depression 469 Find the measure of each angle to the nearest tenth of a degree. (Lesson 8-4) 29. cos A = 0.6717 30. sin B = 0.5127 1 , find tan B. (Lesson 8-4) 32. If cos B = _ " 4 ! Find x and y. (Lesson 8-3) 33. 31. tan C = 2.1758 # 34. 35. y x y 14 12 30˚ 45˚ y 60˚ 20 x x 36. LANDSCAPING Paulo is designing two gardens shaped like similar triangles. One garden has a perimeter of 53.5 feet, and the longest side is 25 feet. He wants the second garden to have a perimeter of 32.1 feet. Find the length of the longest side of this garden. (Lesson 7-5) 37. MODEL AIRPLANES A twin-engine airplane used for medium-range flights has a length of 78 meters and a wingspan of 90 meters. If a scale model is made with a wingspan of 36 centimeters, find its length. (Lesson 6-2) 38. Copy and complete the flow proof. (Lesson 4-6) Given: ∠5 ∠6 −− −− FR GS F 5 G 3 4 1 Prove: ∠4 ∠3 2 X R Proof: 6 S ⬔5 ⬔6 Given FR GS 䉭FXR 䉭GXS Given ? a. Vert. ⬔s are . b. c. ? d. ? e. ? f. ? ? Determine the truth value of the following statement for each set of conditions. If you have a fever, then you are sick. (Lesson 2-3) 39. You do not have a fever, and you are sick. 40. You have a fever, and you are not sick. 41. You do not have a fever, and you are not sick. 42. You have a fever, and you are sick. PREREQUISITE SKILL Solve each proportion. (Lesson 7-1) 35 x 43. _ =_ 6 42 3 5 _ 44. _ x= 45 470 Chapter 8 Right Triangles and Trigonometry 12 24 45. _ =_ x 17 x 24 46. _ =_ 36 15 8-6 The Law of Sines Main Ideas • Use the Law of Sines to solve triangles. • Solve problems by using the Law of Sines. New Vocabulary Law of Sines solving a triangle The Statue of Liberty was designed by Frederic-Auguste Bartholdi between 1865 and 1875. Copper sheets were hammered and fastened to an interior skeletal framework, which was designed by Alexandre-Gustave Eiffel. The skeleton is 94 feet high and composed of wrought iron bars. These bars are arranged in triangular shapes, many of which are not right triangles. The Law of Sines In trigonometry, the Law of Sines can be used to find missing parts of triangles that are not right triangles. 8.8 Obtuse Angles There are also values for sin A, cos A, and tan A, when A ≥ 90°. Values of the ratios for these angles will be found using the trigonometric functions on your calculator. Law of Sines Let ABC be any triangle with a, b, and c representing the measures of the sides opposite the angles with measures A, B, and C, respectively. Then sin C sin A sin B _ = _ = _. a PROOF b a A c b C B c Law of Sines ABC is a triangle with an altitude −− from C that intersects AB at D. Let h −−− represent the measure of CD. Since ADC and BDC are right triangles, we can find sin A and sin B. A h sin A = _ h a D B a sin B = h Cross products b sin A = a sin B sin A sin B _ =_ a b h sin B = _ a Definition of sine b b sin A = h C b Substitution Divide each side by ab. The proof can be completed by using a similar technique with the sin A sin C sin B sin C _ _ other altitudes to show that _ =_ a = c and c . b Lesson 8-6 The Law of Sines Rohan/Stone/Getty Images 471 EXAMPLE Use the Law of Sines Given measures of ABC, find the indicated measure. Round angle measures to the nearest degree and side measures to the nearest tenth. a. If m∠A = 37, m∠B = 68, and a = 3, find b. Use the Law of Sines to write a proportion. sin A sin B _ =_ a b sin 37° sin 68° _=_ 3 b Rounding If you round before the final answer, your results may differ from results in which rounding was not done until the final answer. Law of Sines m∠A = 37, a = 3, m∠B = 68 b sin 37° = 3 sin 68° Cross products 3 sin 68° b=_ Divide each side by sin 37°. sin 37° b ≈ 4.6 Use a calculator. b. If b = 17, c = 14, and m∠B = 92, find m∠C. Write a proportion relating ∠B, ∠C, b, and c. sin B sin C _ =_ c b sin 92° sin C _=_ 17 14 Law of Sines m∠B = 92, b = 17, c = 14 14 sin 92° = 17 sin C Cross products 14 sin 92° _ = sin C 17 14 sin 92° sin-1 _ =C 17 ) ( 55° ≈ C Divide each side by 17. Solve for C. Use a calculator. So, m∠C ≈ 55. 1A. If m∠B = 32, m∠C = 51, c = 12, find a. 1B. If a = 22, b = 18, m∠A = 25, find m∠B. The Law of Sines can be used to solve a triangle. Solving a triangle means finding the measures of all of the angles and sides of a triangle. EXAMPLE Look Back To review the Angle Sum Theorem, see Lesson 4-2. Solve Triangles a. Solve ABC if m∠A = 33, m∠B = 47, and b = 14. Round angle measures to the nearest degree and side measures to the nearest tenth. We know the measures of two angles of the triangle. Use the Angle Sum Theorem to find m∠C. 472 Chapter 8 Right Triangles and Trigonometry C a b A c Extra Examples at geometryonline.com B An Equivalent Proportion The Law of Sines may also be written as c a b _ =_ = _. sin A sin B m∠A + m∠B + m∠C = 180 33 + 47 + m∠C = 180 80 + m∠C = 180 m∠C = 100 Angle Sum Theorem m∠A = 33, m∠B = 47 Add. Subtract 80 from each side. sin B Since we know m∠B and b, use proportions involving _ . b sin C You may wish to use this form when finding the length of a side. To find a: To find c: sin B sin A _ =_ a b sin 47° sin 33° _ _ = a 14 a sin 47° = 14 sin 33° Law of Sines sin B sin C _ =_ Substitute. sin 47° sin 100° _ =_ b 14 Divide each side by sin 47°. sin 47° a ≈ 10.4 c c sin 47° = 14 sin 100° Cross products 14 sin 33° a=_ c 14 sin 100° c=_ sin 47° c ≈ 18.9 Use a calculator. Therefore, m∠C = 100, a ≈ 10.4, and c ≈ 18.9. b. Solve ABC if m∠C = 98, b = 14, and c = 20. Round angle measures to the nearest degree and side measures to the nearest tenth. sin B sin C _ =_ Law of Sines c b sin B sin 98° _ _ = 20 14 m∠C = 98, b = 14, and c = 20 20 sin B = 14 sin 98° Cross products 14 sin 98° sin B = _ 20 ( Divide each side by 20. ) 14 sin 98° B = sin-1 _ Solve for B. 20 B ≈ 44° m∠A + m∠B + m∠C = 180 m∠A + 44 + 98 = 180 m∠A + 142 = 180 m∠A = 38 sin A sin C _ =_ a c sin 38° sin 98° _ =_ a 20 Use a calculator. Angle Sum Theorem m∠B = 44 and m∠C = 98 Add. Subtract 142 from each side. Law of Sines m∠A = 38, m∠C = 98, and c = 20 20 sin 38° = a sin 98° Cross products 20 sin 38° _ =a Divide each side by sin 98°. sin 98° 12.4 ≈ a Use a calculator. Therefore, A ≈ 38°, B ≈ 44°, and a ≈ 12.4. Interactive Lab geometryonline.com Find the missing angles and sides of PQR. Round angle measures to the nearest degree and side measures to the nearest tenth. 2A. m∠R = 66, m∠Q = 59, p = 72 2B. p = 32, r = 11, m∠P = 105 Lesson 8-6 The Law of Sines 473 Use the Law of Sines to Solve Problems The Law of Sines is very useful in solving direct and indirect measurement applications. Indirect Measurement ENGINEERING When the angle of elevation to the Sun is 62°, a telephone pole tilted at an angle of 7° from the vertical casts a shadow 30 feet long on the ground. Find the length of the telephone pole to the nearest tenth of a foot. 3 Draw a diagram. 0 −− −−− Draw SD ⊥ GD. Then find m∠GDP and m∠GPD. POLE m∠GDP = 90 - 7 or 83 m∠GPD + 62 + 83 = 180 or m∠GPD = 35 ' FOOTSHADOW $ Since you know the measures of two angles of the triangle, m∠GDP and −−− m∠GPD, and the length of a side opposite one of the angles (GD is opposite ∠GPD) you can use the Law of Sines to find the length of the pole. GD PD _ =_ Law of Sines sin ∠DGP sin ∠GPD 30 PD _ =_ m∠DGP = 62, m∠GPD = 35, and GD = 30 sin 62° sin 35° PD sin 35° = 30 sin 62° Cross products 30 sin 62° PD = _ Divide each side by sin 35°. sin 35° PD ≈ 46.2 Use a calculator. The telephone pole is about 46.2 feet long. 3. AVIATION Two radar stations that are 35 miles apart located a plane at the same time. The first station indicated that the position of the plane made an angle of 37° with the line between the stations. The second station indicated that it made an angle of 54° with the same line. How far is each station from the plane? Personal Tutor at geometryonline.com Law of Sines Case 2 of the Law of Sines can lead to two different triangles. This is called the ambiguous case of the Law of Sines. Law of Sines The Law of Sines can be used to solve a triangle in the following cases. Case 1 You know the measures of two angles and any side of a triangle. (AAS or ASA) Case 2 You know the measures of two sides and an angle opposite one of these sides of the triangle. (SSA) 474 Chapter 8 Right Triangles and Trigonometry Example 1 (p. 472) Find each measure using the given measures of XYZ. Round angle measures to the nearest degree and side measures to the nearest tenth. 1. If x = 3, m∠X = 37, and m∠Y = 68, find y. 2. If y = 12.1, m∠X = 57, and m∠Z = 72, find x. 3. If y = 7, z = 11, and m∠Z = 37, find m∠Y. 4. If y = 17, z = 14, and m∠Y = 92, find m∠Z. 5. SURVEYING To find the distance between two points A and B that are on opposite sides of a river, a surveyor measures the distance to point C on the same side of the river as point A. The distance from A to C is 240 feet. He then measures the angle across from A to B as 62° and measures the angle across from C to B as 55°. Find the distance from A to B. Example 2 (p. 472) Example 3 (p. 474) C A B Solve each PQR described below. Round angle measures to the nearest degree and side measures to the nearest tenth. 6. m∠P = 33, m∠R = 58, q = 22 7. p = 28, q = 22, m∠P = 120 8. m∠P = 50, m∠Q = 65, p = 12 9. q = 17.2, r = 9.8, m∠Q = 110.7 10. m∠P = 49, m∠R = 57, p = 8 11. m∠P = 40, m∠Q = 60, r = 20 12. Find the perimeter of parallelogram ABCD to the nearest tenth. A B 6 D HOMEWORK HELP For See Exercises Examples 13–18 1 19–26 2 27, 28 3 32° 88° C Find each measure using the given measures of KLM. Round angle measures to the nearest degree and side measures to the nearest tenth. 13. If k = 3.2, m∠L = 52, and m∠K = 70, find . 14. If m = 10.5, k = 18.2, and m∠K = 73, find m∠M. 15. If k = 10, m = 4.8, and m∠K = 96, find m∠M. 16. If m∠M = 59, = 8.3, and m = 14.8, find m∠L. 17. If m∠L = 45, m∠M = 72, and = 22, find k. 18. If m∠M = 61, m∠K = 31, and m = 5.4, find . Solve each WXY described below. Round measures to the nearest tenth. 19. m∠Y = 71, y = 7.4, m∠X = 41 20. x = 10.3, y = 23.7, m∠Y = 96 21. m∠X = 25, m∠W = 52, y = 15.6 22. m∠Y = 112, x = 20, y = 56 23. m∠W = 38, m∠Y = 115, w = 8.5 24. m∠W = 36, m∠Y = 62, w = 3.1 25. w = 30, y = 9.5, m∠W = 107 26. x = 16, w = 21, m∠W = 88 Lesson 8-6 The Law of Sines 475 IN 27. TELEVISIONS To gain better reception on his antique TV, Mr. Ramirez positioned the two antennae 13 inches apart with an angle between them of approximately 82°. If one antenna is 5 inches long, about how long is the other antenna? IN XIN 28. REAL ESTATE A house is built on a triangular plot of land. Two sides of the plot are 160 feet long, and they meet at an angle of 85°. If a fence is to be placed along the perimeter of the property, how much fencing material is needed? 85˚ 160 ft 160 ft 29. An isosceles triangle has a base of 46 centimeters and a vertex angle of 44°. Find the perimeter. 30. Find the perimeter of quadrilateral ABCD to the nearest tenth. A B 28˚ 12 40˚ C D 31. SURVEYING Maria Lopez is a surveyor who must determine the distance across a section of the Rio Grande Gorge in New Mexico. On one side of the ridge, she measures the angle formed by the edge of the ridge and the line of sight to a tree on the other side of the ridge. She then walks along the ridge 315 feet, passing the tree and measures the angle formed by the edge of the ridge and the new line of sight to the same tree. If the first angle is 80° and the second angle is 85°, find the distance across the gorge. EXTRA PRACTICE See pages 816, 835. Self-Check Quiz at geometryonline.com H.O.T. Problems HIKING For Exercises 32 and 33, use the following information. Kayla, Jenna, and Paige are hiking at a state park and they get separated. Kayla and Jenna are 120 feet apart. Paige sends up a signal. Jenna turns 95° in the direction of the signal and Kayla rotates 60°. 32. To the nearest foot, how far apart are Kayla and Paige? 33. To the nearest foot, how far apart are Jenna and Paige? 34. FIND THE ERROR Makayla and Felipe are trying to find d in DEF. Who is correct? Explain your reasoning. Makayla d sin 59° = _ 12 F Felipe 12 73˚ sin 59° _ _ = sin 48° d 12 D f d 48˚ E 35. OPEN ENDED Draw an acute triangle and label the measures of two angles and the length of one side. Explain how to solve the triangle. 36. CHALLENGE Does the Law of Sines apply to the acute angles of a right triangle? Explain your answer. 37. Writing in Math Refer to the information on the Statue of Liberty on page 471. Describe how triangles are used in structural support. 476 Chapter 8 Right Triangles and Trigonometry 38. Soledad is looking at the top of a 150-foot tall Ferris wheel at an angle of 75°. 39. REVIEW Which inequality best describes the graph below? 5 4 3 2 1 sin 75° ≈ 0.97 cos 75° ≈ 0.26 FT ⫺3⫺2⫺1 O tan 75° ≈ 3.73 1 2 3 4 5x ⫺2 ⫺3 FT If she is 5 feet tall, how far is Soledad from the Ferris wheel? y F y ≥ -x + 2 G y≤x+2 A 15.0 ft C 75.8 ft H y ≥ -3x + 2 B 38.9 ft D 541.1 ft J y ≤ 3x + 2 ARCHITECTURE For Exercises 40 and 41, use the following information. Mr. Martinez is an architect who designs houses so that the windows receive minimum Sun in the summer and maximum Sun in the winter. For Columbus, Ohio, the angle of elevation of the Sun at noon on the longest day is 73.5° and on the shortest day is 26.5°. Suppose a house is designed with a south-facing window that is 6 feet tall. The top of the window is to be installed 1 foot below the overhang. (Lesson 8-5) x 1 ft window 6 ft high angle of elevation of sun 40. How long should the architect make the overhang so that the window gets no direct sunlight at noon on the longest day? 41. Using the overhang from Exercise 40, how much of the window will get direct sunlight at noon on the shortest day? Use JKL to find sin J, cos J, tan J, sin L, cos L, and tan L. Express each ratio as a fraction and as a decimal to the nearest hundredth. (Lesson 8-4) 42. j = 8, k = 17, l = 15 43. j = 20, k = 29, l = 21 44. j = 12, k = 24, l = 12 √3 45. j = 7 √2, k = 14, l = 7 √2 J K L _ 2 2 2 PREREQUISITE SKILL Evaluate c - a - b for the given values of a, b, and c. (Page 780) -2ab 46. a = 7, b = 8, c = 10 47. a = 4, b = 9, c = 6 48. a = 5, b = 8, c = 10 49. a = 16, b = 4, c = 13 50. a = 3, b = 10, c = 9 51. a = 5, b = 7, c = 11 Lesson 8-6 The Law of Sines 477 Geometry Software Lab EXTEND 8-6 The Ambiguous Case of the Law of Sines In Lesson 8-6, you learned that you could solve a triangle using the Law of Sines if you know the measures of two angles and any side of the triangle (AAS or ASA). You can also solve a triangle by the Law of Sines if you know the measures of two sides and an angle opposite one of the sides (SSA). When you use SSA to solve a triangle, and the given angle is acute, sometimes it is possible to find two different triangles. You can use The Geometer’s Sketchpad to explore this case, called the ambiguous case, of the Law of Sines. ACTIVITY . Construct a circle and AC Step 1 Construct AB whose center is B so that it intersects AC at two points. Then, construct any −− radius BD. −− −− Step 2 Find the measures of BD, AB, and ∠A. Step 3 Use the rotate tool to move D so that it lies on one of the intersection points of . In ABD, find the circle B and AC −−− measures of ∠ABD, ∠BDA, and AD. Step 4 Using the rotate tool, move D to the other intersection point of circle B . and AC Step 5 Note the measures of ∠ABD, −−− ∠BDA, and AD in ABD. Ambiguous Case BD = 3.50 cm m∠ABD = 97.44˚ AB = 5.79 cm m∠ADB = 53.48˚ m∠BAC = 29.26˚ AD = 7.15 cm Ambiguous Case BD = 3.50 cm m∠ABD = 24.81˚ AD = 3.00 cm AB = 5.79 cm m∠ADB = 125.82˚ m∠BAC = 29.26˚ B B A C D C A D ANALYZE THE RESULTS 1. Which measures are the same in both triangles? −− −− 2. Repeat the activity using different measures for ∠A, BD, and AB. How do the results compare to the earlier results? 3. Compare your results with those of your classmates. How do the results compare? 4. What would have to be true about circle B in order for there to be one unique solution? Test your conjecture by repeating the activity. −− −− 5. Is it possible, given the measures of BD, AB, and ∠A, to have no solution? Test your conjecture and explain. 478 Chapter 8 Right Triangles and Trigonometry 8-7 Main Ideas • Use the Law of Cosines to solve triangles. • Solve problems by using the Law of Cosines. The Law of Cosines German architect Ludwig Mies van der Rohe entered the design at the right in the Friedrichstrasses Skyscraper Competition in Berlin in 1921. The skyscraper was to be built on a triangular plot of land. In order to maximize space, the design called for three towers in a triangular shape. However, the skyscraper was never built. New Vocabulary Law of Cosines The Law of Cosines Suppose you know the lengths of the sides of the triangular building and want to solve the triangle. The Law of Cosines allows us to solve a triangle when the Law of Sines cannot be used. 8.9 Law of Cosines Let ABC be any triangle with a, b, and c representing the measures of sides opposite angles A, B, and C, respectively. Then the following equations are true. Side and Angle a2 = b2 + c2 - 2bc cos A Note that the letter of the side length on the left-hand side of each equation corresponds to the angle measure used with the cosine. b2 = a2 + c2 C b A a B c - 2ac cos B c2 = a2 + b2 - 2ab cos C The Law of Cosines can be used to find missing measures in a triangle if you know the measures of two sides and their included angle. EXAMPLE A Two Sides and the Included Angle Find a if c = 8, b = 10, and m∠A = 60. Use the Law of Cosines since the measures of two sides and the included are known. a2 = b2 + c2 - 2bc cos A C a 8 B Law of Cosines a2 = 102 + 82 - 2(10)(8) cos 60° b = 10, c = 8, and m∠A = 60 a2 Simplify. = 164 - 160 cos 60° 60˚ 10 a = √ 164 - 160 cos 60° Take the square root of each side. a ≈ 9.2 Use a calculator. 1. In DEF, e = 19, f = 28, and m∠D = 49. Find d. Extra Examples at geometryonline.com Digital Image ©The Museum of Modern Art/Licensed by SCALA/Art Resource, NY Lesson 8-7 The Law of Cosines 479 You can also use the Law of Cosines to find the measures of angles of a triangle when you know the measures of the three sides. EXAMPLE Three Sides Q 23 18 Find m∠R. R r2 = q2 + s2 - 2qs cos R Law of Cosines 232 = 372 + 182 - 2(37)(18) cos R r = 23, q = 37, s = 18 529 = 1693 - 1332 cos R Simplify. -1164 = -1332 cos R Subtract 1693 from each side. -1164 _ = cos R Divide each side by -1332. -1332 ( 1332 ) S 37 1164 R = cos-1 _ Solve for R. R ≈ 29.1° Use a calculator. 2. In TVW, v = 18, t = 24, and w = 30. Find m∠W. Use the Law of Cosines to Solve Problems Most problems can be solved using more than one method. Choosing the most efficient way to solve a problem is sometimes not obvious. When solving right triangles, you can use sine, cosine, or tangent ratios. When solving other triangles, you can use the Law of Sines or the Law of Cosines. You must decide how to solve each problem depending on the given information. Solving a Triangle To solve Right triangle Any triangle EXAMPLE Given Begin by using two legs tangent leg and hypotenuse sine or cosine angle and hypotenuse sine or cosine angle and a leg sine, cosine, or tangent two angles and any side Law of Sines two sides and the angle opposite one of them Law of Sines two sides and the included angle Law of Cosines three sides Law of Cosines Select a Strategy Solve KLM. Round angle measures to the nearest degree and side measures to the nearest tenth. We do not know whether KLM is a right triangle, so we must use the Law of Cosines or the Law of K Sines. We know the measures of two sides and the included angle. This is SAS, so use the Law of Cosines. 480 Chapter 8 Right Triangles and Trigonometry L k 14 51˚ 18 M Law of Cosines If you use the Law of Cosines to find another measure, your answer may differ slightly from one found using the Law of Sines. This is due to rounding. k2 = 2 + m2 - 2m cos K Law of Cosines k2 + - 2(18)(14) cos 51° k = √ 182 + 142 - 2(18)(14) cos 51° = 18, m = 14, and m∠K = 51 k ≈ 14.2 Use a calculator. = 182 142 Take the square root of each side. Next, we can find m∠L or m∠M. If we decide to find m∠L, we can use either the Law of Sines or the Law of Cosines to find this value. In this case, we will use the Law of Sines. sin L sin K _ =_ Law of Sines k sin L sin 51° _ _ ≈ 18 14.2 = 18, k ≈ 14.2, and m∠K = 51 14.2 sin L ≈ 18 sin 51° Cross products 18 sin 51° sin L ≈ _ 14.2 18 sin 51° L ≈ sin-1 _ 14.2 ( Divide each side by 14.2. ) L ≈ 80° Take the inverse sine of each side. Use a calculator. Use the Angle Sum Theorem to find m∠M. m∠K + m∠L + m∠M = 180 51 + 80 + m∠M ≈ 180 m∠M ≈ 49 Angle Sum Theorem m∠K = 51 and m∠L ≈ 80 Subtract 131 from each side. Therefore, k ≈ 14.2, m∠K ≈ 80, and m∠M ≈ 49. 3. Solve XYZ for x = 10, y = 11, and z = 12. Personal Tutor at ca.geometryonline.com Use the Law of Cosines REAL ESTATE Ms. Jenkins is buying some property that is shaped like quadrilateral ABCD. Find the perimeter of the property. C 200 ft B Use the Pythagorean Theorem to find BD in ABD. (AB)2 + (AD)2 = (BD)2 1802 + 2402 = (BD)2 90,000 = (BD)2 300 = BD 60˚ Pythagorean Theorem 180 ft AB = 180, AD = 240 Simplify. A Take the square root of each side. D 240 ft Next, use the Law of Cosines to find CD in BCD. (CD)2 = (BC)2 + (BD)2 - 2(BC)(BD) cos ∠CBD Law of Cosines (CD)2 = 2002 + 3002 - 2(200)(300) cos 60° BC = 200, BD = 300, m∠CBD = 60 (CD)2 = 130,000 - 120,000 cos 60° Simplify. CD = √ 130,000 - 120,000 cos 60° Take the square root of each side. CD ≈ 264.6 Use a calculator. The perimeter is 180 + 200 + 264.6 + 240 or about 884.6 feet. Lesson 8-7 The Law of Cosines 481 K 4. ARCHITECTURE An architect is designing a playground in the shape of a quadrilateral. Find the perimeter of the playground to the nearest tenth. 15 m 13 32˚ J Example 1 (p. 479) (p. 480) 2. b = 107, c = 94, m∠D = 105 In RST, given the lengths of the sides, find the measure of the stated angle to the nearest degree. 3. r = 33, s = 65, t = 56; m∠S Example 3 (p. 480) 4. r = 2.2, s = 1.3, t = 1.6; m∠R Solve each triangle using the given information. Round angle measures to the nearest degree and side measures to the nearest tenth. 5. XYZ: x = 5, y = 10, z = 13 Example 4 (p. 481) HOMEWORK HELP For See Exercises Examples 8–11 1 12–15 2 16–22, 3 25–32 23, 24 4 M 18 m In BCD, given the following measures, find the measure of the missing side. 2 , d = 5, m∠B = 45 1. c = √ Example 2 L 6. JKL: j = 20, = 24, m∠K = 47 7. BASKETBALL Josh and Brian are playing basketball. Josh passes the ball to Brian, who takes a shot. Josh is 12 feet from the hoop and 10 feet from Brian. The angle formed by the hoop, Josh, and Brian is 34°. Find the distance Brian is from the hoop. *OSH "RIAN In TUV, given the following measures, find the measure of the missing side. 8. t = 9.1, v = 8.3, m∠U = 32 10. u = 11, v = 17, m∠T = 105 9. t = 11, u = 17, m∠V = 78 11. v = 11, u = 17, m∠T = 59 In EFG, given the lengths of the sides, find the measure of the stated angle to the nearest degree. 12. e = 9.1, f = 8.3, g = 16.7; m∠F 14. e = 325, f = 198, g = 208; m∠F 13. e = 14, f = 19, g = 32; m∠E 15. e = 21.9, f = 18.9, g = 10; m∠G Solve each triangle using the given information. Round angle measures to the nearest degree and side measures to the nearest tenth. 16. 17. G 8 40˚ F 18. Q C 10 11 g H 482 Chapter 8 Right Triangles and Trigonometry 11 M B 38˚ P 10 18 p 15 D Solve each triangle using the given information. Round angle measures to the nearest degree and side measures to the nearest tenth. 19. ABC: m∠A = 42, m∠C = 77, c = 6 20. ABC: a = 10.3, b = 9.5, m∠C = 37 21. ABC: a = 15, b = 19, c = 28 22. ABC: m∠A = 53, m∠C = 28, c = 14.9 23. KITES Beth is building a kite like the one at the −− −− right. If AB is 5 feet long, BC is 8 feet long, and −− 2 BD is 7_ feet long, find the measures of the angle 3 between the short sides and the angle between the long sides to the nearest degree. A B D C Real-World Link The Swissôtel in Chicago, Illinois, is built in the shape of a triangular prism. The lengths of the sides of the triangle are 180 feet, 186 feet, and 174 feet. Source: Swissôtel 24. BUILDINGS Refer to the information at the left. Find the measures of the angles of the triangular building to the nearest tenth. Solve each LMN described below. Round measures to the nearest tenth. 25. m = 44, = 54, m∠L = 23 26. m∠M = 46, m∠L = 55, n = 16 27. m = 256, = 423, n = 288 28. m∠M = 55, = 6.3, n = 6.7 29. m∠M = 27, = 5, n = 10 30. n = 17, m = 20, = 14 31. = 14, m = 15, n = 16 32. m∠L = 51, = 40, n = 35 33. In quadrilateral ABCD, 34. In quadrilateral PQRS, PQ = 721, AC = 188, BD = 214, m∠BPC = 70, QR = 547, RS = 593, PS = 756, and −− and P is the midpoint of AC and m∠P = 58. Find QS, m∠PQS, and −− BD. Find the perimeter of ABCD. m∠R. A Q P B P D C R S 35. SOCCER Carlos and Adam are playing soccer. Carlos is standing 40 feet from one post of the goal and 50 feet from the other post. Adam is standing 30 feet from one post of the goal and 22 feet from the other post. If the goal is 24 feet wide, which player has a greater angle to make a shot on goal? Carlos 40 ft 50 ft 24 ft 30 ft 22 ft Adam 36. Each side of regular hexagon ABCDEF is 18 feet long. What is the length of the −− diagonal BD? Explain your reasoning. " # ! $ & % Lesson 8-7 The Law of Cosines Pierre Burnaugh/PhotoEdit 483 EXTRA PRACTICE See pages 816, 835. Self-Check Quiz at geometryonline.com 37. PROOF Justify each statement for the derivation of the Law of Cosines. −−− Given: AD is an altitude of ABC. Prove: c2 = a2 + b2 - 2ab cos C " ! C H $ X AX # A Proof: Statement Reasons a. c2 = (a - x)2 + h2 a. ? b. c2 = a2 - 2ax + x2 + h2 b. ? c. x2 + h2 = b2 c. ? d. c2 = a2 - 2ax + b2 d. ? x e. cos C = _ e. ? f. b cos C = x f. ? g. c2 = a2 - 2a(b cos C) + b2 g. ? h. c2 = a2 + b2 - 2ab cos C h. ? b H.O.T. Problems B 38. OPEN ENDED Draw and label one acute and one obtuse triangle, illustrating when you can use the Law of Cosines to find the missing measures. 39. REASONING Find a counterexample for the following statement. The Law of Cosines can be used to find the length of a missing side in any triangle. 40. CHALLENGE Graph A(-6, -8), B(10, -4), C(6, 8), and D(5, 11) on the coordinate plane. Find the measure of interior angle ABC and the measure of exterior angle DCA. 14 12 10 8 6 4 2 ⫺8⫺6⫺4⫺2 O ⫺4 ⫺6 ⫺8 A (⫺6, ⫺8)⫺10 y D (5, 11) C (6, 8) x 2 4 6 8 10 12 B (10, ⫺4) 41. Which One Doesn’t Belong? Analyze the four terms and determine which does not belong with the others. 42. Pythagorean triple Pythagorean Theorem Law of Cosines cosine Writing in Math Refer to the information about the Friedrichstrasses Skyscraper Competition on page 479. Describe how triangles were used in van der Rohe’s design. Explain why the Law of Cosines could not be used to solve the triangle. 484 Chapter 8 Right Triangles and Trigonometry 43. In the figure below, cos B = 0.8. " 45. REVIEW The scatter plot shows the responses of swim coaches to a survey about the hours of swim team practice and the number of team wins. ! 4EAM7INS # −− What is the length of AB? A 12.8 B 16.8 0RACTICEH C 20.0 D 28.8 44. REVIEW Which of the following shows 2x2 - 24xy - 72y2 factored completely? F (2x - 18y)(x + 4y) G 2(x - 6y)(x + 6y) H (2x - 8y)(x - 9) J 2(x - 6y)(x + 18y) Which statement best describes the relationship between the two quantities? A As the number of practice hours increases, the number of team wins increases. B As the number of practice hours increases, the number of team wins decreases. C As the number of practice hours increases, the number of team wins at first decreases, then increases. D There is no relationship between the number of practice hours and the number of team wins. Find each measure using the given measures from XYZ. Round angle measure to the nearest degree and side measure to the nearest tenth. (Lesson 8-6) 46. If y = 4.7, m∠X = 22, and m∠Y = 49, find x. 47. If y = 10, x = 14, and m∠X = 50, find m∠Y. 48. SURVEYING A surveyor is 100 meters from a building and finds that the angle of elevation to the top of the building is 23°. If the surveyor’s eye level is 1.55 meters above the ground, find the height of the building. (Lesson 8-5) −− −− A For Exercises 49–51, determine whether AB CD. (Lesson 7-4) 49. AC = 8.4, BD = 6.3, DE = 4.5, and CE = 6 50. AC = 7, BD = 10.5, BE = 22.5, and AE = 15 51. AB = 8, AE = 9, CD = 4, and CE = 4 C B D E COORDINATE GEOMETRY The vertices of XYZ are X(8, 0), Y(-4, 8), and Z(0, 12). Find the coordinates of the points of concurrency of XYZ to the nearest tenth. (Lesson 5-1) 52. orthocenter 53. centroid 54. circumcenter Lesson 8-7 The Law of Cosines 485 CH APTER 8 Study Guide and Review Download Vocabulary Review from geometryonline.com 3IGHT5RIANGLES Key Vocabulary Be sure the following Key Concepts are noted in your Foldable. angle of depression (p. 465) angle of elevation (p. 464) cosine (p. 456) geometric mean (p. 432) Pythagorean triple (p. 443) Key Concepts Geometric Mean sine (p. 456) solving a triangle (p. 472) tangent (p. 456) trigonometric ratio (p. 456) trigonometry (p. 456) (Lesson 8-1) • For two positive numbers a and b, the geometric mean is the positive number x where the proportion a : x = x : b is true. This proportion a x can be written using fractions as _x = _ or with b . cross products as x2 = ab or x = √ab Pythagorean Theorem (Lesson 8-2) • In a right triangle, the sum of the squares of the measures of the legs equals the square of the hypotenuse. Special Right Triangles (Lesson 8-3) • The measures of the sides of a 45°–45°–90° triangle are x, x, and x √2. • The measures of the sides of a 30°–60°–90° triangle are x, x √3, and 2x. Trigonometry (Lesson 8-4) • Trigonometric Ratios: opposite leg hypotenuse sin A = _ Vocabulary Check State whether each sentence is true or false. If false, replace the underlined word or number to make a true sentence. 1. To solve a triangle means to find the measures of all its sides and angles. 2. The Law of Sines can be applied if you know the measures of two sides and an angle opposite one of these sides of the triangle. 3. In any triangle, the sum of the squares of the measures of the legs equals the square of the measure of the hypotenuse. 4. An angle of depression is the angle between the line of sight and the horizontal when an observer looks upward. adjacent leg cos A = _ hypotenuse opposite leg adjacent leg tan A = _ Laws of Sines and Cosines (Lessons 8-6 and 8-7) Let ABC be any triangle with a, b, and c representing the measures of the sides opposite the angles with measures A, B, and C, respectively. 5. The geometric mean between two numbers is the positive square root of their product. 6. A 30°-60°-90° triangle is isosceles. sin A _ sin C sin B • Law of Sines: _ =_ a = c 7. Looking at a city while flying in a plane is an example that uses an angle of elevation. • Law of Cosines: 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 8. The numbers 3, 4, and 5 form a Pythagorean identity. b 486 Chapter 8 Right Triangles and Trigonometry Vocabulary Review at geometryonline.com Lesson-by-Lesson Review 8-1 Geometric Mean (pp. 432–438) Example 1 Find the geometric mean between 10 and 30. Find the geometric mean between each pair of numbers. 10. 4 and 81 9. 4 and 16 11. 20 and 35 13. In PQR, PS = 8, and QS = 14. Find RS. Q P Cross products Simplify. Example 2 Find NG in TGR. R The Pythagorean Theorem and Its Converse The measure of the altitude is the geometric mean between the measures of the two hypotenuse segments. GN TN _ =_ RN GN GN 2 _ _ = GN 4 8 = (GN)2 √ 8 or 2 √ 2 = GN T 2 N 4 G R Definition of geometric mean TN = 2, RN = 4 Cross products Take the square root of each side. (pp. 440–446) Example 3 Use JKL to find a. Find x. 15. 15 x2 = 300 30 x = √ 300 or 10 √ 3 14. INDIRECT MEASUREMENT To estimate the height of the Space Needle in Seattle, Washington, James held a book up to his eyes so that the top and bottom of the building were in line with the bottom edge and binding of the cover. If James’ eye level is 6 feet from the ground and he is standing 60 feet from the tower, how tall is the tower? 8-2 Definition of geometric mean x 12. 18 and 44 S 10 x _ =_ 16. x 13 17 J 5 17 x 13 a 20 17. FARMING A farmer wishes to create a maze in his corn field. He cuts a path 625 feet across the diagonal of the rectangular field. Did the farmer create two right triangles? Explain. K a 2 + (LK) 2 = (JL) 2 2 2 a + 8 = 13 2 2 a + 64 = 169 a 2 = 105 FT 8 L Pythagorean Theorem LK = 8 and JL = 13 Simplify. Subtract 64 from each side. a = √ 105 Take the square root of each side. a ≈ 10.2 Use a calculator. FT Chapter 8 Study Guide and Review 487 CH A PT ER 8 8-3 Study Guide and Review Special Right Triangles Find x and y. 18. y 45˚ (pp. 448–454) Example 4 Find x. −− The shorter leg, XZ, of XYZ is half the measure of the −− hypotenuse XY. 30˚ 19. x 9 x y 60˚ 6 x 30˚ z b Y Example 5 Find x. a The hypotenuse of a 45°-45°-90° triangle is √ 2 times the length of a leg. Q 4 P x √2 = 4 45˚ x x R 4 x=_ √ 2 4 x=_ √ 2 Trigonometry 26 x 22. ORIGAMI To create a bird, Michelle first folded a square piece of origami paper along one of the diagonals. If the diagonal measured 8 centimeters, find the length of one side of the square. 8-4 60˚ 1 Therefore, XZ = _ (26) Z 2 or 13. The longer leg is √3 times the measure of the shorter leg. So, x = 13 √3. For Exercises 20 and 21, use the figure. 20. If y = 18, find z y 60˚ and a. 21. If x = 14, find a, z, b, and y. X • √ 2 _ or 2 √ 2 √ 2 (pp. 456–462) Use FGH to find sin F, cos F, tan F, sin G, cos G, and tan G. Express each ratio as a fraction and as a decimal to the nearest hundredth. G 23. f = 9, g = 12, h = 15 h 24. f = 7, g = 24, h = 25 25. f = 9, g = 40, h = 41 Example 6 Find sin A, cos A, and tan A. Express as a fraction and as a decimal. B 13 A f 12 5 C opposite leg adjacent leg cos A = _ hypotenuse hypotenuse BC AC =_ =_ AB AB 5 12 or about 0.38 =_ or =_ 13 13 sin A = _ F g H 26. SPACE FLIGHT A space shuttle is directed towards the Moon but drifts 0.8° from its calculated path. If the distance from Earth to the Moon is 240,000 miles, how far has the space shuttle drifted from its path when it reaches the Moon? 488 Chapter 8 Right Triangles and Trigonometry about 0.92 opposite leg tan A = _ adjacent leg BC =_ AC _ = 5 or about 0.42 12 Mixed Problem Solving For mixed problem-solving practice, see page 835. 8-5 Angles of Elevation and Depression (pp. 464–470) Determine the angles of elevation or depression in each situation. 27. Upon takeoff, an airplane must clear a 60-foot pole at the end of a runway 500 yards long. 28. An escalator descends 100 feet for each horizontal distance of 240 feet. 29. A hot-air balloon ascends 50 feet for every 1000 feet traveled horizontally. 30. EAGLES An eagle, 1350 feet in the air, notices a rabbit on the ground. If the horizontal distance between the eagle and the rabbit is 700 feet, at what angle of depression must the eagle swoop down to catch the rabbit and fly in a straight path? 8-6 The Law of Sines Example 7 The ramp of a loading dock measures 12 feet and has a height of 3 feet. What is the angle of elevation? Make a drawing. B 12 3 x˚ A C Let x represent m∠BAC. opposite leg hypotenuse BC sin x° = _ sin x = _ AB 3 sin x° = _ 12 BC = 3 and AB = 12 ( 12 ) 3 x = sin⁻¹ _ Find the inverse. x ≈ 14.5 Use a calculator. The angle of elevation for the ramp is about 14.5°. (pp. 471–477) Find each measure using the given measures of FGH. Round angle measures to the nearest degree and side measures to the nearest tenth. Example 8 Find x if y = 15. Round to the nearest tenth. Y 32. Find m∠H if h = 10.5, g = 13, and m∠G = 65. 33. GARDENING Elena is planning a triangular garden. She wants to build a fence around the garden to keep out the deer. The length of one side of the garden is 26 feet. If the angles at the end of this side are 78° and 44°, find the length of fence needed to enclose the garden. 61˚ z 31. Find f if g = 16, m∠G = 48, and m∠F = 82. X 32˚ y x Z To find x and z, use proportions involving sin Y and y. sin Y sin X _ =_ y x sin 32° sin 61° _ _ = x 15 x sin 61° = 15 sin 32° Law of Sines Substitute. Cross Products 15 sin 32° x=_ Divide. x ≈ 9.1 Use a calculator. sin 61° Chapter 8 Study Guide and Review 489 CH A PT ER 8 8-7 Study Guide and Review The Law of Cosines (pp. 479–485) In XYZ, given the following measures, find the measures of the missing side. Example 9 Find a. B 34. x = 7.6, y = 5.4, m∠Z = 51 19 35. x = 21, m∠Y = 73, z = 16 54˚ a Solve each triangle using the given information. Round angle measures to the nearest degree and side measures to the nearest tenth. a 2 = b 2 + c 2 - 2bc cos A Law of Cosines 36. c = 18, b = 13, m∠A = 64 a 2 = 23 2 + 19 2 - 2(23)(19) cos 54° A 23 C 37. b = 5.2, m∠C = 53, c = 6.7 38. ART Adelina is creating a piece of art that is in the shape of a parallelogram. Its dimensions are 35 inches by 28 inches and one angle is 80°. Find the lengths of both diagonals. 490 Chapter 8 Right Triangles and Trigonometry a 2 = 890 - 874 cos 54° b = 23, c = 19, and m∠A = 54 Simplify. a = √ 890 - 874 cos 54° Take the square root of each side. a ≈ 19.4 Use a calculator. CH A PT ER 8 Practice Test Find the geometric mean between each pair of numbers. 1. 7 and 63 2. 6 and 24 3. 10 and 50 Find the missing measures. 4. 6 x 5. 7 x 21. CIVIL ENGINEERING A section of freeway has a steady incline of 10°. If the horizontal distance from the beginning of the incline to the end is 5 miles, how high does the incline reach? 13 5 6. 7. 9 x 9 12 y 9. y x 60˚ 8 45˚ 16 y˚ 12 x x 17 C _ 12 B _ 12 D _ 13 B 21 A 16 15 C 24. Solve DEF. Solve each triangle. Round each angle measure to the nearest degree and each side measure to the nearest tenth. 17. a = 15, b = 17, m∠C = 45 18. a = 12.2, b = 10.9, m∠B = 48 19. a = 19, b = 23.2, c = 21 Chapter Test at geometryonline.com 12 5 23. COMMUNICATIONS To secure a 500-foot radio tower against high winds, guy wires are attached to the tower 5 feet from the top. The wires form a 15° angle with the tower. Find the distance from the centerline of the tower to the anchor point of the wires. Find each measure using the given measures from FGH. Round to the nearest tenth. 13. Find g if m∠F = 59, f = 13, and m∠G = 71. 14. Find m∠H if m∠F = 52, f = 10, and h = 12.5. 15. Find f if g = 15, h = 13, and m∠F = 48. 16. Find h if f = 13.7, g = 16.8, and m∠H = 71. £Î 5 A _ 12 8 Use the figure to find each trigonometric ratio. Express answers as a fraction. 10. cos B 11. tan A 12. sin A 22. MULTIPLE CHOICE Find tan X. 19 x 8. 20. TRAVEL From an airplane, Janara looked down to see a city. If she looked down at an angle of 9° and the airplane was half a mile above the ground, what was the horizontal distance to the city? D 12 E 82˚ 8 F 25. MULTIPLE CHOICE The top of the Boone Island Lighthouse in Boone Island, Maine, is 137 feet above sea level. The angle of depression from the light on the top of the tower to a passing ferry is 37°. How many feet from the foot of the lighthouse is the ferry? F 181.8 ft H 109.4 ft G 171.5 ft J 103.2 ft Chapter 8 Practice Test 491 CH A PT ER 8 Standardized Test Practice Cumulative, Chapters 1–8 Read each question. Then fill in the correct answer on the answer document provided by your teacher or on a sheet of paper. 3. A detour has been set up on the interstate due to a gas leak. The diagram below shows the detour route. How many extra miles will drivers have to travel due to the detour? 1. A diagram from a proof of the Pythagorean Theorem is pictured below. Which statement would be used in the proof of the Pythagorean Theorem? a St at e 18 miles Ro gh wa y ut e Hi 24 miles b te Intersta A B C D c A The area of the larger square equals (a + b) 2. B The area of the inner square is equal to half of the area of the larger square. C The area of the larger square is equal to the sum of the areas of the smaller square and the four congruent triangles. 12 miles 30 miles 42 miles 80 miles and 8. 4. A right triangle has legs of length √39 What is the length of the hypotenuse? F 10 H 11 G √103 J 4 √26 D The four right triangles are similar. 2. In the figure below, if tan x = __43 , what are cos x and sin x? Question 4 If a standardized test question involves trigonometric ratios, draw a diagram that represents the problem. Use a calculator (if allowed) or the table of trigonometric values provided to help you find the answer. −− −− 5. Given: BD AE 4 # x 3 F G H J 3 4 cos x = __ , sin x = __ 5 4 3 5 , sin x = __ cos x = __ 4 4 3 4 , sin x = __ cos x = __ 5 5 3 5 , sin x = __ cos x = __ 5 4 492 Chapter 8 Right Triangles and Trigonometry " ! $ % What theorem or postulate can be used to prove ACE ∼ BCD? A SSS B SAS C ASA D AA Standardized Test Practice at geometryonline.com Preparing for Standardized Tests For test-taking strategies and more practice, see pages 841–856. −− 6. In ABC, D is the midpoint of AB, and E is 10. Rhombus ABCD is shown. −− the midpoint of AC. Which pair of triangles can be aestablished to be congruent to prove −− −− that AC bisects BD? A 2 1 E 4 D " ! # A ABD and CBD C B ACD and ACB 3 C AEB and BEC B G ABC ∼ ADE 11. What is the shortest side of quadrilateral DEFG? AE AD = ___ J ____ DB $ D AEB and CED Which of the following is not true? −− −− F ∠1 ∠4 H DE BC EC H G 65˚ 7. If the sum of the measures of the interior angles of a polygon is 900, how many sides does the polygon have? A5 B 7 55˚ C 8 D 10 12. An extension ladder leans against the side of a house while gutters are being cleaned. The base of the ladder is 12 feet from the house, and the top of the ladder rests 16 feet up the side of the house. a. Draw a figure representing this situation. What is the length of the ladder? b. For safety, a ladder should have a climbing angle of no less than 75°. Is the climbing angle of this ladder safe? c. If not, what distance from the house should the ladder be placed so that it still rests 16 feet up the side of the house at a 75° climbing angle and to what new length will the ladder need to be adjusted? 40 m Main Street East 9. ALGEBRA Find (x 2 + 2x - 24) ÷ (x - 4). J x+6 −−− H DG −− J DE Record your answer on a sheet of paper. Show your work. 46 m G x+8 E Pre-AP Ma in S tre et W est H x-6 F 55˚ 85˚ D triangular traffic median on Main Street to provide more green space in the downtown area. The planner builds a model so that the section of the median facing Main Street East measures 20 centimeters. What is the perimeter, in centimeters, of the model of the traffic median? F x-8 35˚ −− F GF −− G FE 8. GRIDDABLE A city planner designs a 23 m % NEED EXTRA HELP? If You Missed Question... 1 2 3 4 5 6 7 8 9 10 11 12 Go to Lesson or Page... 8-2 8-4 8-2 8-2 7-3 7-4 6-1 7-5 794 6-3 5-3 8-7 Chapter 8 Standardized Test Practice 493

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