• Lessons 7-1, 7-2, and 7-3 Solve problems using the geometric mean, the Pythagorean Theorem, and its converse. • Lessons 7-4 and 7-5 Use trigonometric ratios to solve right triangle problems. • Lessons 7-6 and 7-7 Solve triangles using the Law of Sines and the Law of Cosines. Trigonometry is used to find the measures of the sides and angles of triangles. These ratios are frequently used in real-world applications such as architecture, aviation, and surveying. You will learn how surveyors use trigonometry in Lesson 7-6. 340 Chapter 7 Right Triangles and Trigonometry Bob Daemmrich/The Image Works Key Vocabulary • • • • • geometric mean (p. 342) Pythagorean triple (p. 352) trigonometric ratio (p. 364) Law of Sines (p. 377) Law of Cosines (p. 385) Prerequisite Skills To be successful in this chapter, you’ll need to master these skills and be able to apply them in problem-solving situations. Review these skills before beginning Chapter 7. For Lesson 7-1 Proportions Solve each proportion. Round to the nearest hundredth, if necessary. (For review, see Lesson 6-1.) c 8 2. 3 12 1. 4 5 a f 10 e 6 3. 20 3 5 4 6 1 4. 3 For Lesson 7-2 y z Pythagorean Theorem Find the measure of the hypotenuse of each right triangle having legs with the given measures. Round to the nearest hundredth, if necessary. (For review, see Lesson 1-3.) 5. 5 and 12 6. 6 and 8 7. 15 and 15 8. 14 and 27 For Lessons 7-3 and 7-4 Radical Expressions Simplify each expression. (For review, see pages 744 and 745.) 9. 8 10. 102 52 7 12. 2 11. 392 362 For Lessons 7-5 through 7-7 Angle Sum Theorem Find x. (For review, see Lesson 4-2.) 13. 14. 38˚ 15. (2x + 21)˚ 40˚ 44˚ x˚ x˚ 155˚ B x˚ A C Right Triangles and Trigonometry Make this Foldable to help you organize your notes. Begin with seven sheets of grid paper. Fold each sheet along the diagonal from the corner of one end to 2.5 inches away from the corner of the other end. Staple Staple the sheets in three places. 2.5 in. Stack Stack the sheets, and fold the rectangular part in half. Label Label each sheet with a lesson number, and the rectangular part with the chapter title. Ch. 7 Right Triangles Fold 7-1 Reading and Writing As you read and study the chapter, write notes, define terms, and solve problems in your Foldable. Chapter 7 Right Triangles and Trigonometry 341 Geometric Mean • Find the geometric mean between two numbers. • Solve problems involving relationships between parts of a right triangle and the altitude to its hypotenuse. can the geometric mean be used to view paintings? Vocabulary • geometric mean 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. GEOMETRIC MEAN The geometric mean between two numbers is the positive square root of their product. Geometric Mean Study Tip Means and Extremes In the equation x2 ab, the two x’s in x2 represent the means, and a and b represent the extremes of the proportion. 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 be written using fractions as x a or with cross products as x2 ab or x ab . x b Example 1 Geometric Mean Find the geometric mean between each pair of numbers. a. 4 and 9 Let x represent the geometric mean. 4 x x 9 Definition of geometric mean x2 36 Cross products x 36 Take the positive square root of each side. x6 Simplify. b. 6 and 15 x 6 x 15 x2 90 Cross products x 90 Take the positive square root of each side. x 310 Simplify. x 9.5 Use a calculator. 342 Chapter 7 Right Triangles and Trigonometry Robert Brenner/PhotoEdit Definition of geometric mean ALTITUDE OF A TRIANGLE Consider right triangle XYZ 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. Z X W Y 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. Explore the relationships among the three right triangles formed. Z Think and Discuss 1. Find the measures of X, XZY, Y, XWZ, XZW, YWZ, and YZW. 2. What is the relationship between the measures of X and YZW? What is the relationship between the measures of Y and XZW ? Y W X 3. Drag point Z to another position. Describe the relationship between the measures of X and YZW and between the measures of Y and XZW. 4. Make a conjecture about XYZ, XZW, and ZYW. Study Tip 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. The results of the Geometry Software Investigation suggest the following theorem. Theorem 7.1 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. Y X Z W Example: XYZ XWY YWZ You will prove this theorem in Exercise 45. By Theorem 7.1, since XWY YWZ, the corresponding sides are proportional. XW YW YW ZW XW Thus, . Notice that and Z W are segments of the hypotenuse of the largest triangle. Theorem 7.2 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 this theorem in Exercise 46. www.geometryonline.com/extra_examples/sol Lesson 7-1 Geometric Mean 343 Example 2 Altitude and Segments of the Hypotenuse In PQR, RS 3 and QS 14. Find PS. Let x PS. RS PS PS QS x 3 14 x Study Tip Square Roots Since these numbers represent measures, you can ignore the negative square root value. P Q S R RS 3, QS 14, and PS x x2 42 Cross products x 42 Take the positive square root of each side. x 6.5 Use a calculator. PS is about 6.5. Ratios in right triangles can be used to solve problems. Example 3 Altitude and Length of the Hypotenuse ARCHITECTURE Mr. Martinez is designing a walkway that must pass over an elevated train. To find the height of the elevated train, 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 distance from the street to the top of the train. Round to the nearest tenth. Z X be the altitude Draw a diagram. Let Y drawn from the right angle of WYZ. WX YX YX ZX 5.5 8.75 8.75 ZX WX 5.5 and YX 8.75 5.5ZX 76.5625 Y Cross products X 5.5 ft ZX 13.9 Divide each side by 5.5. 8.75 ft W Mr. Martinez estimates that the elevated train is 5.5 13.9 or about 19.4 feet high. The altitude to the hypotenuse of a right triangle determines another relationship between the segments. Theorem 7.3 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 measures of the hypotenuse and the segment of the hypotenuse adjacent to that leg. Y X XZ XY ZY XZ Example: and XY XW YZ WZ You will prove Theorem 7.3 in Exercise 47. 344 Chapter 7 Right Triangles and Trigonometry W Z Example 4 Hypotenuse and Segment of Hypotenuse Find x and y in PQR. Q and R P Q are legs of right triangle PQR. Use Theorem 7.3 to write a proportion for each leg and then solve. Study Tip Simplifying Radicals Remember that 12 4 3. Since 4 2, 12 23. For more practice simplifying radicals, see pages 744 and 745. Concept Check PR PQ PQ PS y 6 y 2 y2 12 PS 2, PQ y, PR 6 Cross products R 4 x S 2 PR RQ RQ SR 6 x x 4 P Q y RS 4, RQ x, PR 6 x2 24 Cross products y 12 Take the square root. x 24 Take the square root. y 23 Simplify. x 26 Simplify. y 3.5 Use a calculator. x 4.9 Use a calculator. 1. OPEN ENDED Find two numbers whose geometric mean is 12. 2. 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 7.3. 3. FIND THE ERROR RST is a right isosceles triangle. Holly and Ian are finding the measure of altitude S U . Ian Holly RU SU = UT SU RS SU = SU RT x 9.9 = 14 x S 7 x = x 7 x 2 = 138.6 9.9 x x2 = 49 x = 138.6 R 7 x= 7 U 7 T x ≈ 11.8 Who is correct? Explain your reasoning. Guided Practice Find the geometric mean between each pair of numbers. 4. 9 and 4 5. 36 and 49 6. 6 and 8 7. 22 and 32 Find the measure of the altitude drawn to the hypotenuse. 8. A 9. E 2 D 6 C B G H 16 12 F Lesson 7-1 Geometric Mean 345 Find x and y. 10. x B Application 11. C 8 D C y y B A 3 2 √3 x D A 2 12. DANCES Khaliah 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, Khaliah held a book up to her eyes so that the top and bottom of the wall were in line with the top edge and binding of the cover. If Khaliah’s eye level is 5 feet off the ground and she is standing 12 feet from the wall, how high is the wall? Practice and Apply For Exercises See Examples 13–20 21–26 27–32 1 2 3, 4 Extra Practice Find the geometric mean between each pair of numbers. 13. 5 and 6 14. 24 and 25 15. 45 and 80 83 63 18. and 3 17. and 1 5 5 16. 28 and 1372 22 52 19. and 5 6 13 5 20. and 6 7 Find the measure of the altitude drawn to the hypotenuse. 21. B 22. F 23. J 5 See page 766. 7 8 M 16 D H 12 9 A E C 24. 25. Q L G X 26. Z V 2W 13 10 P 21 S 7 U R T 28. x y x z x 8 z 31. x y z 10 4 15 y 5 y 30. N 29. 6 z 3 2.5 Y Find x, y, and z. 27. 8 K 12 32. z y 36 6x 12 x z 8 y x 346 Chapter 7 Right Triangles and Trigonometry The geometric mean and one extreme are given. Find the other extreme. 33. 17 is the geometric mean between a and b. Find b if a 7. 34. 12 is the geometric mean between x and y. Find x if y 3. Determine whether each statement is always, sometimes, or never true. 35. The geometric mean for consecutive positive integers is the average of the two numbers. 36. The geometric mean for two perfect squares is a positive integer. 37. The geometric mean for two positive integers is another integer. 38. The measure of the altitude of a triangle is the geometric mean between the measures of the segments of the side it intersects. 39. BIOLOGY The shape of the shell of a chambered nautilus can be modeled by a geometric mean. Consider the sequence OC of segments O A , O B , , O D , O E , O F , OG , O H , O I, and O J. 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.) You can use geometric mean and the Quadratic Formula to discover the golden mean. Visit www.geometry online.com/webquest to continue work on your WebQuest project. G C J O B F H D A E I 40. RESEARCH Refer to the information at the left. Use the Internet or other resource to write a brief description of the golden ratio. 41. CONSTRUCTION In the United States, most building codes limit the steepness of 4 the slope of a roof to , as shown at the 3 right. A builder wants to put a support brace from point C perpendicular to AP . Find the length of the brace. P 5 yd 4 yd A SOCCER For Exercises 42 and 43, refer to the graphic. 42. Find the geometric mean between the number of players from Indiana and North Carolina. 43. Are there two schools whose geometric mean is the same as the geometric mean between UCLA and Clemson? If so, which schools? 3 yd C B USA TODAY Snapshots® Bruins bring skills to MLS Universities producing the most players in Major League Soccer this season: 15 UCLA 10 Indiana 9 Virginia 44. CRITICAL THINKING Find the exact value of DE, given AD 12 and BD 4. C North Carolina 7 Washington 7 Clemson 6 Source: MLS A D www.geometryonline.com/self_check_quiz/sol E B By Ellen J. Horrow and Adrienne Lewis, USA TODAY Lesson 7-1 Geometric Mean 347 Kaz Chiba/PhotoDisc Write the specified type proof for each theorem. 45. two-column proof 46. paragraph proof of 47. two-column proof of Theorem 7.1 of Theorem 7.2 of Theorem 7.3 PROOF 48. WRITING IN MATH Answer the question that was posed at the beginning of the lesson. How can the geometric mean be used to view paintings? Include the following in your answer: • an explanation of what happens when you are too far or too close to a painting, and • an explanation of how the curator of a museum would determine where to place roping in front of paintings on display. SOL/EOC Practice Standardized Test Practice 49. Find x and y. A 4 and 6 C 3.6 and 6.4 B D 8 cm 2.5 and 7.5 3 and 7 6 cm y cm 10 cm 50. ALGEBRA Solve 5x2 405 1125. A 15 B 12 C 43 D 4 x cm Maintain Your Skills Mixed Review Find the first three iterations of each expression. (Lesson 6-6) 51. x 3, where x initially equals 12 52. 3x 2, where x initially equals 4 2 54. 2(x 3), where x initially equals 1 53. x 2, where x initially equals 3 55. 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 6-5) Use the Exterior Angle Inequality Theorem to list all angles that satisfy the stated condition. (Lesson 5-2) 56. all angles with a measure less than m8 57. all angles with a measure greater than m1 58. all angles with a measure less than m7 59. all angles with a measure greater than m6 7 8 3 1 2 5 6 Write an equation in slope-intercept form for the line that satisfies the given conditions. (Lesson 3-4) 60. m 2, y-intercept 4 61. x-intercept is 2, y-intercept 8 62. passes through (2, 6) and (1, 0) 63. m 4, passes through (2, 3) Getting Ready for the Next Lesson PREREQUISITE SKILL Use the Pythagorean Theorem to find the length of the hypotenuse of each right triangle. (To review using the Pythagorean Theorem, see Lesson 1-4.) 64. 65. 3 cm 5 ft 12 ft 66. 5 in. 4 cm 348 Chapter 7 Right Triangles and Trigonometry 3 in. 4 A Preview of Lesson 7-2 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. Use square pieces of patty paper and algebra. Then you too can discover this relationship among the measures of the sides of a right triangle. Activity Use paper folding to develop the Pythagorean Theorem. a 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 Copy these measures on the other sides in the order shown b ab at the right. Fold the paper to divide the square into four sections. Label the a a2 area of each section. a On another sheet of patty paper, mark the same lengths a and b on the sides in the different pattern shown at the right. b Use your straightedge and pencil to connect the marks as shown at the right. Let c represent the length of each hypotenuse. b2 b a ab b a b 1 2 ab 1 2 ab c c c2 2 c2 c c Step 5 Label the area of each section, which is 1 ab for each triangle and b c c c for the square. 1 2 ab c 1 2 ab Step 6 Place the squares side by side and color the corresponding regions that have the 1 1 same area. For example, ab ab ab. 2 a 2 1 2 ab b ab 1 2 ab b2 b c2 = a b a a2 ab 1 2 ab 1 2 ab The parts that are not shaded tell us that a2 b2 c2. Model 1. Use a ruler to find actual measures for a, b, and c. Do these measures measures confirm that a2 b2 c2? 2. Repeat the activity with different a and b values. What do you notice? Analyze the model 3. Explain why the drawing at the right is an illustration of the Pythagorean Theorem. Investigating Slope-Intercept Form 349 Geometry Activity The Pythagorean Theorem 349 The Pythagorean Theorem and Its Converse Virginia SOL Standard G.7 The student will solve practical problems involving right triangles by using the Pythagorean Theorem …. Solutions will be expressed in radical form or as decimal approximations. • Use the Pythagorean Theorem. • Use the converse of the Pythagorean Theorem. are right triangles used to build suspension bridges? Vocabulary • Pythagorean triple The Talmadge Memorial Bridge over the Savannah River 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. Study Tip Look Back THE PYTHAGOREAN THEOREM In Lesson 1-3, you used the Pythagorean To review finding the hypotenuse of a right triangle, see Lesson 1-3. 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. Theorem 7.4 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. Symbols: a2 b2 B c A a C b c2 The geometric mean can be used to prove the Pythagorean Theorem. Proof Pythagorean Theorem Given: ABC with right angle at C Prove: a2 b2 c2 C a Proof: Draw right triangle ABC so C is the right angle. . Let AB c, Then draw the altitude from C to AB AC b, BC a, AD x, DB y, and CD h. B b h y D Two geometric means now exist. c a a y and c b b x a2 cy and b2 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). 350 Chapter 7 Right Triangles and Trigonometry x c A Example 1 Find the Length of the Hypotenuse LONGITUDE AND LATITUDE NASA Dryden is located at about 117 degrees longitude and 34 degrees latitude. NASA Ames 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. Maps 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 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 34 c 5.8 c Pythagorean Theorem a 5, b 3 Simplify. Add. Take the square root of each side. Use a calculator. The degree distance between NASA Dryden and NASA Ames is about 5.8 degrees. Example 2 Find the Length of a Leg Find x. (XY)2 (YZ)2 (XZ)2 7 2 x2 142 49 x2 196 x2 147 x 147 x 73 x 12.1 X Pythagorean Theorem XY 7, XZ 14 Simplify. Subtract 49 from each side. 14 in. 7 in. Y x in. Z Take the square root of each side. Simplify. Use a calculator. 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. Theorem 7.5 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 b a C You will prove this theorem in Exercise 38. www.geometryonline.com/extra_examples/sol Lesson 7-2 The Pythagorean Theorem and Its Converse 351 Example 3 Verify a Triangle is a Right Triangle Study Tip 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. COORDINATE GEOMETRY Verify that PQR is a right triangle. Use the Distance Formula to determine the lengths of the sides. (–3 3)2 (6 2 )2 PQ Q (–3, 6) y R (5, 5) P (3, 2) x1 3, y1 2, x2 –3, y2 6 2 (–6) 42 Subtract. 52 Simplify. O x 2 QR [5 (–3)] (5 6 )2 x1 –3, y1 6, x2 5, y2 5 82 (–1) 2 Subtract. 65 Simplify. PR (5 3 )2 (5 2)2 x1 = 3, y1 = 2, x2 = 5, y2 = 5 Subtract. 13 Simplify. 22 32 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 Converse of the Pythagorean Theorem 13 65 2 52 13 65 65 65 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. A Pythagorean triple is three whole numbers that satisfy the equation a2 b2 c2, where c is the greatest number. One common Pythagorean triple is 3-4-5, in which the sides of a right triangle are in the ratio 3 : 4 : 5. If the measures of the sides of any right triangle are whole numbers, the measures form a Pythagorean triple. Example 4 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 15 and 8. a2 b2 82 152 64 225 289 c2 Pythagorean Theorem 2 16 a 8, b 15, c 16 256 Simplify. 256 Add. Since 289 256, segments with these measures cannot form a right triangle. Therefore, they do not form a Pythagorean triple. 352 Chapter 7 Right Triangles and Trigonometry b. 20, 48, and 52 a2 b2 c2 202 482 522 400 2304 2704 2704 2704 If you cannot quickly identify the greatest number, use a calculator to find decimal values for each number and compare. a 20, b 48, c 52 Simplify. Add. These segments form the sides of a right triangle since they satisfy the Pythagorean Theorem. The measures are whole numbers and form a Pythagorean triple. Study Tip Comparing Numbers Pythagorean Theorem 3 , 6 , and 3 c. 5 5 5 a2 b2 c2 Pythagorean Theorem 53 56 35 2 2 2 5 3 6 9 25 25 25 5 5 Simplify. 9 9 25 25 9 3 6 3 a , b , c Add. 9 Since , segments with these measures form a right triangle. However, 25 25 the three numbers are not whole numbers. Therefore, they do not form a Pythagorean triple. Concept Check 1. FIND THE ERROR Maria and Colin are determining whether 5-12-13 is a Pythagorean triple. Colin Maria a2 + b2 = c2 ? 132 + 52 = 122 ? 169 + 25 = 144 193 ≠ 144 no a2 + b2 = c2 ? 5 2 + 12 2 = 13 2 ? 25 + 144 = 169 169 = 169 yes Who is correct? Explain your reasoning. 2. Explain why a Pythagorean triple can represent the measures of the sides of a right triangle. 3. OPEN ENDED Draw a pair of similar right triangles. List the corresponding sides, the corresponding angles, and the scale factor. Are the measures of the sides of each triangle a Pythagorean triple? Guided Practice Find x. 4. 6 5. 10 x 4 7 x 5 7 6. x 37.5 20 Lesson 7-2 The Pythagorean Theorem and Its Converse 353 7. 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. 9. 40 , 20, 21 8. 15, 36, 39 Application 10. 44 , 8, 108 11. COMPUTERS Computer monitors are usually measured along the diagonal of the screen. A 19-inch monitor has a diagonal that measures 19 inches. If the height of the screen is 11.5 inches, how wide is the screen? 19 in. 11.5 in. Practice and Apply For Exercises See Examples 14, 15 12, 13, 16, 17 18–21 22–29 1 2 3 4 Extra Practice Find x. 12. 8 13. 8 14. 8 x 4 28 x 14 15. 20 16. 40 17. 33 See page 767. x x x 25 25 15 32 x COORDINATE GEOMETRY Determine whether QRS is a right triangle for the given vertices. Explain. 18. Q(1, 0), R(1, 6), S(9, 0) 19. Q(3, 2), R(0, 6), S(6, 6) 20. Q(–4, 6), R(2, 11), S(4, –1) 21. 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. 22. 8, 15, 17 23. 7, 24, 25 24. 20, 21, 31 25. 37, 12, 34 1 1 74 26. , , 5 7 35 5 3 , 2 , 3 27. 2 3 36 3 4 28. , , 1 5 5 6 8 10 29. , , 7 7 7 For Exercises 30–35, use the table of Pythagorean triples. a b c 30. Copy and complete the table. 5 12 13 31. A primitive Pythagorean triple is a Pythagorean triple 10 24 ? with no common factors except 1. Name any primitive 15 ? 39 Pythagorean triples contained in the table. ? 48 52 32. Describe the pattern that relates these sets of Pythagorean triples. 33. These Pythagorean triples are called a family. Why do you think this is? 34. Are the triangles described by a family of Pythagorean triples similar? Explain. 35. For each Pythagorean triple, find two triples in the same family. a. 8, 15, 17 b. 9, 40, 41 c. 7, 24, 25 354 Chapter 7 Right Triangles and Trigonometry Aaron Haupt GEOGRAPHY For Exercises 36 and 37, use the following information. Denver is located at about 105 degrees longitude and 40 degrees latitude. San Francisco is located at about 122 degrees longitude and 38 degrees latitude. Las Vegas is located at about 115 degrees longitude and 36 degrees latitude. Using the lines of longitude and latitude, find each degree distance. 125° 120° 115° 110° 105° 100° 40° San Francisco Denver Las Vegas 35° 30° 36. San Francisco to Denver 37. Las Vegas to Denver 38. PROOF Write a paragraph proof of Theorem 7.5. 39. PROOF Use the Pythagorean Theorem and the figure at the right to prove the Distance Formula. y A(x1, y1) d C (x1, y2) B (x2, y2) x O 40. 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? 41. SAILING The mast of a sailboat is supported by wires called shrouds. What is the total length of wire needed to form these shrouds? 16 ft x 2 ft 12 ft 42. 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. 12 ft 3 ft 26 ft 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. Online Research For information about a career in the military, visit: www.geometryonline. com/careers x shrouds 9 ft 43. 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? H 44. CRITICAL THINKING The figure at the right is a rectangular prism with AB 8, BC 6, and BF 8, and M is the midpoint of BD. Find BD and HM. How are EM, FM, and GM related to HM? F D A www.geometryonline.com/self_check_quiz/sol G E M C B Lesson 7-2 The Pythagorean Theorem and Its Converse 355 Answer the question that was posed at the beginning of the lesson. How are right triangles used to build suspension bridges? Include the following in your answer: • the locations of the right triangles, and • an explanation of which parts of the right triangle are formed by the cables. 45. WRITING IN MATH SOL/EOC Practice Standardized Test Practice Graphing Calculator 46. In the figure, if AE 10, what is the value of h? A 6 B 8 C 10 D 12 2 2 47. ALGEBRA If x 36 (9 x) , then find x. A 6 B no solution C 2.5 D 10 y E (8, h) A O C x PROGRAMMING For Exercises 48 and 49, use the following information. The TI-83 Plus program uses a procedure for finding Pythagorean triples PROGRAM: PYTHTRIP that was developed by Euclid around :For (X, 2, 6) :Disp B,A,C 320 B.C. Run the program to generate a :For (Y, 1, 5) :Else list of Pythagorean triples. :If X > Y :Disp A,B,C 48. List all the members of the 3-4-5 :Then :End family that are generated by the 2 2 :int (X Y + 0.5)➞A :End program. :2XY➞B :Pause 49. A geometry student made the 2 2 :int (X +Y +0.5)➞C :Disp ” ” conjecture that if three whole :If A > B :End numbers are a Pythagorean triple, :Then :End then their product is divisible by 60. Does this conjecture hold true for :Stop each triple that is produced by the program? Maintain Your Skills Mixed Review Find the geometric mean between each pair of numbers. (Lesson 7-1) 50. 3 and 12 51. 9 and 12 52. 11 and 7 53. 6 and 9 54. 2 and 7 55. 2 and 5 Find the value of each expression. Then use that value as the next x in the expression. Repeat the process and describe your observations. (Lesson 6-6) 56. 2x , where x initially equals 5 57. 3x, where x initially equals 1 58. x , where x initially equals 4 1 59. , where x initially equals 4 1 2 x 60. Determine whether the sides of a triangle could have the lengths 12, 13, and 25. Explain. (Lesson 5-4) Getting Ready for the Next Lesson PREREQUISITE SKILL Simplify each expression by rationalizing the denominator. (To review simplifying radical expressions, see pages 744 and 745.) 61. 7 3 12 66. 3 356 Chapter 7 Right Triangles and Trigonometry 18 62. 2 26 67. 3 14 63. 2 311 64. 3 24 65. 2 15 68. 3 69. 2 8 25 70. 10 Special Right Triangles Virginia SOL Standard G.7 The student will solve practical problems involving right triangles by using … properties of special right triangles, …. Solutions will be expressed in radical form or as decimal approximations. • Use properties of 45°-45°-90° triangles. • Use properties of 30°-60°-90° triangles. is triangle tiling used in wallpaper design? Triangle tiling is the process of laying copies of a single triangle next to each other to fill an area. One type of triangle tiling is wallpaper tiling. There are exactly 17 types of triangle tiles that can be used for wallpaper tiling. 1 2 10 3 11 12 4 5 13 6 14 7 15 8 16 9 17 Tile 4 is made up of two 45°-45°-90° triangles that form a square. This tile is rotated to make the wallpaper design shown at the right. 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. x 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. x d2 x2 x2 d Pythagorean Theorem d2 2x2 Add. d 2x2 Take the positive square root of each side. d 2 Factor. x2 d x2 Simplify. This algebraic proof verifies that the length of the hypotenuse of any 45°-45°-90° triangle is 2 times the length of its leg. The ratio of the sides is 1 : 1 : 2. Theorem 7.6 In a 45°-45°-90° triangle, the length of the hypotenuse is 2 times the length of a leg. n 45˚ n √2 n 45˚ Lesson 7-3 Special Right Triangles 357 Example 1 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. The length of the hypotenuse of one of the triangles is 42. There are four 45°-45°-90° triangles along the diagonal of the square. So, the length of the diagonal of the square is 442 or 162 inches. Example 2 Find the Measure of the Legs Study Tip 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. Find x. The length of the hypotenuse of a 45°-45°-90° triangle is 2 times the length of a leg of the triangle. C xm AB (AC)2 6 x2 AB 6, AC x 6 x 2 2 6 x 2 2 62 x 2 xm 45˚ B 45˚ A 6m Divide each side by 2 . Rationalize the denominator. Multiply. 32 x Divide. 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 K M 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. 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 3x 2 J 30˚ 30˚ 2x 2x a L 60˚ x 60˚ M x K Pythagorean Theorem JM a, LM x, JL 2x Simplify. Subtract x2 from each side. Take the positive square root of each side. a 3 x2 Factor. a x3 Simplify. So, in a 30°-60°-90° triangle, the measures of the sides are x, x3, and 2x. The ratio of the sides is 1 : 3 : 2. 358 Chapter 7 Right Triangles and Trigonometry The relationship of the side measures leads to Theorem 7.7. Theorem 7.7 Study Tip 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 3 times the length of the shorter leg. the longer leg is 30˚ n √3 2n The shorter leg is opposite the 30° angle, and the longer leg is opposite the 60° angle. 60˚ n Example 3 30°-60°-90° Triangles Find AC. C is the longer leg, A BC A B is the shorter leg, and is the hypotenuse. 1 2 B 60˚ 14 in. AB (BC) 1 2 (14) or 7 BC 14 A C AC 3(AB) 3(7) or 73 AB 7 Example 4 Special Triangles in a Coordinate Plane COORDINATE GEOMETRY Triangle PCD is a 30°-60°-90° triangle with right angle C. C D is the longer leg with endpoints C(3, 2) and D(9, 2). Locate point P in Quadrant I. y D lies on a horizontal gridline of Graph C and D. C the coordinate plane. Since PC will be perpendicular to CD CD , it lies on a vertical gridline. Find the length of . CD 9 3 6 CD is the longer leg. P C is the shorter leg. So, CD 3(PC). Use CD to find PC. C (3, 2) x O CD 3(PC) P 6 3(PC) CD 6 6 PC 3 6 3 PC 3 3 Divide each side by 3 . D (9, 2) 60˚ 2x x 30˚ Rationalize the denominator. 63 PC 3 Multiply. 23 PC Simplify. C x √3 6 D Point P has the same x-coordinate as C. P is located 23 units above C. So, the coordinates of P are 3, 2 23 or about (3, 5.46). www.geometryonline.com/extra_examples/sol Lesson 7-3 Special Right Triangles 359 Concept Check Guided Practice 1. OPEN ENDED Draw a 45°-45°-90° triangle. Be sure to label the angles and the sides and to explain how you made the drawing. 2. Explain how to draw a 30°-60°-90° triangle with the shorter leg 2 centimeters long. 3. Write an equation to find the length of a rectangle that has a diagonal twice as long as its width. Find x and y. 4. 5. 3 x 45˚ 6. y 10 x 30˚ 45˚ y 8 x y Find the missing measures. 7. If c 8, find a and b. 8. If b 18, find a and c. B a c 60˚ 30˚ b C A 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. 9. A(8, 0), B(8, 3) 10. A(6, 6), B(2, 6) Application 11. SOFTBALL Find the distance from home plate to second base if the bases are 90 feet apart. 2nd Base 90 ft 3rd Base 90 ft 45˚ 45˚ d 1st Base 45˚ 45˚ 90 ft 90 ft Home Plate Practice and Apply For Exercises See Examples 12, 13, 17, 22, 25 14–16, 18–21, 23, 24 27–31 1, 2 Find x and y. 12. x˚ y 13. 9.6 14. x x y˚ 60˚ 18 17 y 3 4 Extra Practice 15. 16. 17. x 60˚ See page 767. y 12 11 30˚ 60˚ 60˚ y 5 45˚ y x x For Exercises 18 and 19, use the figure at the right. 18. If a 103, find CE and y. 19. If x 73, find a, CE, y, and b. B c y a C 360 Chapter 7 Right Triangles and Trigonometry x E 60˚ 30˚ b A 20. The length of an altitude of an equilateral triangle is 12 feet. Find the length of a side of the triangle. 21. The perimeter of an equilateral triangle is 45 centimeters. Find the length of an altitude of the triangle. 22. The length of a diagonal of a square is 222 millimeters. Find the perimeter of the square. 23. The altitude of an equilateral triangle is 7.4 meters long. Find the perimeter of the triangle. 24. The diagonals of a rectangle are 12 inches long and intersect at an angle of 60°. Find the perimeter of the rectangle. 25. 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. 26. Find x, y, z, and the perimeter of ABCD. z D 8 A C x 30˚ 45˚ 6 y B 27. 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). 28. PGH is a 45°-45°-90° triangle with mP 90. Find the coordinates of P in Quadrant I for G(4, 1) and H(4, 5). 29. 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). 30. PCD is a 30°-60°-90° triangle with mC 30 and hypotenuse CD . Find the coordinates of P for C(2, 5) and D(10, 5) if P lies above CD . 31. If P SR Q , use the figure to find a, b, c, and d. y P (a, b) Q (8√3, d ) 6√3 60˚ S ( 0, 0) 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 45˚ T (c, 0) U (e, 0) R (f, 0) x TRIANGLE TILING For Exercises 32–35, 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. 32. How many 30°-60°-90° triangles are used to create the basic circular pattern? 33. Which angle of the 30°-60°-90° triangle is being rotated to make the basic shape? 34. Explain why there are no gaps in the basic pattern. 35. Use grid paper to cut out 30°-60°-90° triangles. Color the same pattern on each triangle. Create one basic figure that would be part of a wallpaper tiling. Lesson 7-3 Special Right Triangles 361 John Gollings, courtesy Federation Square 36. Find x, y, and z. 37. If BD 83 and mDHB 60, find BH. z 8 H G E y 45˚ F D 60˚ x C A 38. Each triangle in the figure is a 30°-60°-90° triangle. Find x. B 39. In regular hexagon UVWXYZ, each side is 12 centimeters long. Find WY. X W 4 Y V 30˚ 30˚ 30˚ 30˚ x 12 U Z 40. BASEBALL The diagram at the right shows some dimensions of Comiskey Park in Chicago, Illinois. D is a segment from home plate to dead center B field, and A E 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? D C A E 45˚ x 347 ft 45˚ 347 ft B 41. CRITICAL THINKING Given figure ABCD, DC with AB , mB 60, mD 45, BC 8, and AB 24, find the perimeter. A B 60˚ 45˚ D C 42. WRITING IN MATH Answer the question that was posed at the beginning of the lesson. How is triangle tiling used in wallpaper design? Include the following in your answer: • which of the numbered designs contain 30°-60°-90° triangles and which contain 45°-45°-90° triangles, and • a reason why rotations of the basic design left no holes in the completed design. SOL/EOC Practice Standardized Test Practice 43. In the right triangle, what is AB if BC 6? A 12 units B 62 units C 43 units D 23 units A 4x˚ 2x˚ C B a2 44. SHORT RESPONSE For real numbers a and b, where b 0, if a ★ b 2 , then b (3 ★ 4)(5 ★ 3) ? . 362 Chapter 7 Right Triangles and Trigonometry Maintain Your Skills Mixed Review Determine whether each set of measures can be the sides of a right triangle. Then state whether they form a Pythagorean triple. (Lesson 7-2) 45. 3, 4, 5 46. 9, 40, 41 47. 20, 21, 31 48. 20, 48, 52 49. 7, 24, 25 50. 12, 34, 37 Find x, y, and z. (Lesson 7-1) 51. 52. y 53. 10 x 12 4 z z z 8 y x 15 x 5 Write an inequality or equation relating each pair of angles. (Lesson 5-5) 54. mALK, mALN y A 8.5 4.7 4 K N L 6 6 55. mALK, mNLO 4 8.5 56. mOLK, mNLO 6.5 O 57. mKLO, mALN 58. Determine whether JKL with vertices J(3, 2), K(1, 5), and L(4, 4) is congruent to RST with vertices R(6, 6), S(4, 3), and T(1, 4). Explain. (Lesson 4-4) Getting Ready for the Next Lesson PREREQUISITE SKILL Solve each equation. (To review solving equations, see pages 737 and 738.) 59. 5 x 3 x 60. 0.14 7 63. 0.25 64. 9 10 k 9 m 0.8 n 13 g 61. 0.5 62. 0.2 24 65. 0.4 35 66. 0.07 x P ractice Quiz 1 y Lessons 7-1 through 7-3 Find the measure of the altitude drawn to the hypotenuse. (Lesson 7-1) 1. 2. B X 5 9 A 21 7 C Y Z 3. Determine whether ABC with vertices A(2, 1), B(4, 0), and C(5, 7) is a right triangle. Explain. (Lesson 7-2) Find x and y. (Lesson 7-3) 4. y 3 5. y 6 45˚ x www.geometryonline.com/self_check_quiz/sol 30˚ x Lesson 7-3 Special Right Triangles 363 Trigonometry Virginia SOL Standard G.7 The student will solve practical problems involving right triangles by using … right triangle trigonometry. Solutions will be expressed in radical form or as decimal approximations. • Find trigonometric ratios using right triangles. • Solve problems using trigonometric ratios. can surveyors determine angle measures? Vocabulary • • • • • trigonometry trigonometric ratio sine cosine tangent The old surveyor’s telescope shown at right is called a theodolite (thee AH duh lite). It is an optical instrument used to measure angles in surveying, navigation, and meteorology. It consists of a telescope fitted with a level and mounted on a tripod so that it is free to rotate about its vertical and horizontal axes. After measuring 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 Study Tip Reading Math SOH-CAH-TOA is a mnemonic device for learning the ratios for sine, cosine, and tangent using the first letter of each word in the ratios. opp sin A hyp adj hyp cos A opp adj tan A Symbols sine of A measure of leg opposite A measure of hypotenuse BC sin A AB measure of leg opposite B measure of hypotenuse AC sin B AB sine of B cosine of A measure of leg adjacent to A measure of hypotenuse AC cos A AB measure of leg adjacent to B measure of hypotenuse BC cos B AB cosine of B tangent of A measure of leg opposite A measure of leg adjacent to A BC tan A AC measure of leg opposite B tangent of B measure of leg adjacent to B AC tan B BC 364 Chapter 7 Right Triangles and Trigonometry Arthur Thevenart/CORBIS Models B A hypotenuse leg opposite A leg opposite B C B hypotenuse A leg adjacent to A leg adjacent to B C B leg 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. Study Tip Equivalent Ratios Notice that the ratio leg opposite A is the hypotenuse leg adjacent C same as . hypotenuse a Thus, sin A cos C . b Likewise, cos A c sin B . b Trigonometric Ratios A • 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. • Fold the triangle so that there are two segments perpendicular to B A . Label points D, E, F, and G as , A B , B C , AF, AG , shown. Use a ruler to measure AC FG , AD , AE , and DE to the nearest millimeter. A C A ED G F B Analyze C B B C 1. What is true of AED, AGF, and ABC ? 2. Copy the table. Write the ratio of the side lengths for each trigonometric ratio. Then calculate a 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 with the trigonometric ratios. 4. What is true about mA in each triangle? As the Geometry Activity 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 1 Find Sine, Cosine, and Tangent Ratios Find sin R, cos R, tan R, sin S, cos S, and tan S. Express each ratio as a fraction and as a decimal. T S opposite leg hypotenuse sin R adjacent leg hypotenuse cos R ST RS 4 5 or 0.6 or 0.8 opposite leg hypotenuse sin S ST RT or 1.3 adjacent leg hypotenuse 4 3 opposite leg adjacent leg tan S ST RS 4 5 or 0.75 3 5 or 0.8 www.geometryonline.com/extra_examples/sol opposite leg adjacent leg 3 5 cos S R tan R or 0.6 5 RT RS RT RS 3 4 RT ST 3 4 Lesson 7-4 Trigonometry 365 Study Tip 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. Example 2 Use a Calculator to Evaluate Expressions Use a calculator to find each value to the nearest ten thousandth. a. cos 39° KEYSTROKES: COS 39 ENTER cos 39° 0.7771 b. sin 67° SIN 67 ENTER KEYSTROKES: sin 67° 0.9205 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 3 Use Trigonometric Ratios to Find a Length Study Tip 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 was 56°. Find the height of the cliff to the nearest yard. Let x be the height of the cliff in yards. x 97 The tangent of angle A is the same as the slope of the line through the origin that forms an angle of measure A with the positive x-axis. leg opposite tan 56° tan leg adjacent Tangent 97 tan 56° x x yd Multiply each side by 97. Use a calculator to find x. 56˚ KEYSTROKES: 97 TAN 56 ENTER 97 yd 143.8084139 The cliff is about 144 yards high. 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 Read as sin A x A sin1 (x) A equals the inverse sine of x. cos A y tan A z A cos1 (y) A equals the inverse cosine of y. A tan1 (z) A equals the inverse tangent of z. Example 4 Use Trigonometric Ratios to Find an Angle Measure COORDINATE GEOMETRY Find mA 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 366 C(5, 4) A(1, 2) 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 mA. Chapter 7 Right Triangles and Trigonometry O B(6, 2) x Study Tip Calculators The second functions of the SIN , COS , and TAN keys are usually the inverses. Solve AB (6 1)2 (2 2)2 BC (5 6 )2 (4 2)2 25 0 or 5 1 4 or 5 AC (5 1 )2 (4 2)2 16 4 20 or 25 Use the cosine ratio. AC AB cos A leg adjacent cos 25 cos A AC 25 and AB 5 hypotenuse 5 25 A cos1 5 Solve for A. Use a calculator to find mA. KEYSTROKES: 2nd [COS1] 2 2nd [0] 5 ) 5 ENTER mA 26.56505118 The measure of A is about 26.6. Examine Use the sine ratio to check the answer. BC AB leg opposite hypotenuse sin A sin 5 sin A BC 5 and AB 5 5 KEYSTROKES: 2nd [SIN1] 2nd [0 ] 5 5 ENTER ) mA 26.56505118 The answer is correct. Concept Check 1. Explain why trigonometric ratios do not depend on the size of the right triangle. 2. OPEN ENDED Draw a right triangle and label the measures of one acute angle and the measure of the side opposite that angle. Then solve for the remaining measures. 3. Compare and contrast the sine, cosine, and tangent ratios. y x x 4. Explain the difference between tan A y and tan1 A. Guided Practice 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. 5. a 14, b 48, and c 50 6. a 8, b 15, and c 17 B a C c b A Use a calculator to find each value. Round to the nearest ten-thousandth. 7. sin 57° 8. cos 60° 9. cos 33° 10. tan 30° 11. tan 45° 12. sin 85° Lesson 7-4 Trigonometry 367 Find the measure of each angle to the nearest tenth of a degree. 14. sin B 0.6307 13. tan A1.4176 COORDINATE GEOMETRY Find the measure of the angle to the nearest tenth in each right triangle ABC. 15. A in ABC, for A(6, 0), B( 4, 2), and C(0, 6) 16. B in ABC, for A(3, 3), B(7, 5), and C(7, 3) Application 17. 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. Practice and Apply R For Exercises See Examples 18–21, 28–36 22–27 43–48 37–42, 52–54 1 2 3 4 Extra Practice See page 767. Q Use PQR with right angle R to find sin P, cos P, tan P, sin Q, cos Q, and tan Q. Express each ratio as a fraction, and as a decimal to the nearest hundredth. P 18. p 12, q 35, and r 37 19. p 6, q 23, and r 32 3 33 20. p , q , and r 3 21. p 23, q 15 , and r 33 2 2 Use a calculator to find each value. Round to the nearest ten-thousandth. 23. tan 42.8° 24. cos 77° 22. sin 6° 25. sin 85.9° 26. tan 12.7° 27. cos 22.5° Use the figure to find each trigonometric ratio. Express answers as a fraction and as a decimal rounded to the nearest ten-thousandth. 28. sin A 29. tan B 31. sin x° 32. cos x° 34. cos B 35. sin y° C 5√26 A x˚ 5 √26 D B 25 1 30. cos A 33. tan A 36. tan x° Find the measure of each angle to the nearest tenth of a degree. 38. cos C 0.2493 39. tan E 9.4618 37. sin B 0.7245 40. sin A 0.4567 41. cos D 0.1212 42. tan F 0.4279 Find x. Round to the nearest tenth. 44. 43. 45. x x 17 24˚ 62˚ 12 60 19 x˚ 46. 34 x 31˚ 368 Chapter 7 Right Triangles and Trigonometry David R. Frazier/Photo Researchers 47. 17˚ x 6.6 y˚ 48. 18 x˚ 15 49. 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? 3˚ 60 mi 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? 75˚ COORDINATE GEOMETRY Find the measure of each angle to the nearest tenth in each right triangle. 52. J in JCL for J(2, 2), C(2, 2), and L(7, 2) 53. C in BCD for B(1, 5), C(6, 5), and D(1, 2) 54. X in XYZ for X(5, 0), Y(7, 0), and Z(0, 35 ) 55. Find the perimeter of ABC if mA 35, mC 90, and AB 20 inches. Find x and y. Round to the nearest tenth. 56. 57. x˚ x 55˚ 12 24 36 y˚ 58. C y x 24 47˚ 32˚ A B D y ASTRONOMY For Exercises 59 and 60, use the following information. One way to find the distance between the sun and a relatively close star is to determine the angles of sight for the star exactly six months apart. Half the measure formed by these two angles of sight is called the stellar parallax. Distances in space are sometimes measured in astronomical units. An astronomical unit is equal to the average distance between Earth and the sun. Alpha Centauri 0.00021° (stellar parallax) Earth 1 Sun 1 Earth (6 months later) 0.00021° (stellar parallax) Astronomy The stellar parallax is one of several methods of triangulation used to determine the distance of stars from the sun. Another method is trigonometric parallax, which measures the displacement of a nearby star relative to a more distant star. Source: www.infoplease.com 59. Find the distance between Alpha Centauri and the sun. 60. Make a conjecture as to why this method is used only for close stars. 61. CRITICAL THINKING Use the figure at the right to find sin x°. A 10 62. WRITING IN MATH D Answer the question that was posed at the beginning of the lesson. How do surveyors determine angle measures? Include the following in your answer: • where theodolites are used, and • the kind of information one obtains from a theodolite. www.geometryonline.com/self_check_quiz/sol 8 x˚ B 8 10 C Exercise 61 Lesson 7-4 Trigonometry 369 StockTrek/CORBIS Standardized Test Practice SOL/EOC Practice 63. Find cos C. A C B 4 5 D 3 4 5 C 3 5 4 A If x2 152 242 15(24), find x. B 21 C 12 64. ALGEBRA A 20.8 Extending the Lesson B 3 5 9 D Each of the basic trigonometric ratios has a reciprocal ratio. The reciprocals of the sine, cosine, and tangent are called the cosecant, secant, and the cotangent, respectively. A c b B a C Reciprocal Trigonometric Ratio Abbreviation Definition 1 sin A cosecant of A csc A measure of the hypotenuse c measure of the leg opposite A a 1 cos A secant of A sec A measure of the hypotenuse c measure of the leg adjacent A b 1 tan A cotangent of A cot A measure of the leg adjacent A b measure of the leg opposite A a Use ABC to find csc A, sec A, cot A, csc B, sec B, and cot B. Express each ratio as a fraction or as a radical in simplest form. 65. a 3, b 4, and c 5 66. a 12, b 5, and c 13 68. a 22, b 22, and c 4 67. a 4, b 43, and c 8 Maintain Your Skills Mixed Review Find each measure. B (Lesson 7-3) 69. If a 4, find b and c. a 70. If b 3, find a and c. 60˚ c 30˚ C 71. If c 5, find a and b. b A Determine whether each set of measures can be the sides of a right triangle. Then state whether they form a Pythagorean triple. (Lesson 7-2) 72. 4, 5, 6 73. 5, 12, 13 74. 9, 12, 15 75. 8, 12, 16 76. 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 6-1) Getting Ready for the Next Lesson PREREQUISITE SKILL Find each angle measure if h k. (To review angles formed by parallel lines and a transversal, see Lesson 3-2.) 77. m15 78. m7 79. m3 80. m12 81. m11 1 30˚ 82. m4 4 7 370 Chapter 7 Right Triangles and Trigonometry s 6 5 3 2 t 8 9 10 11 12 117˚ 15 14 h k Angles of Elevation and Depression Virginia SOL Standard G.7 The student will solve practical problems involving right triangles by using … right triangle trigonometry. Solutions will be expressed in radical form or as decimal approximations. • Solve problems involving angles of elevation. • Solve problems involving angles of depression. Vocabulary • angle of elevation • angle of depression do airline pilots use angles of elevation and 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. ANGLES OF ELEVATION An A • angle of elevation is the angle between the line of sight and the horizontal when an observer looks upward. B line of sight D angle of elevation C Example 1 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 mDAC. CD AC 450 tan x° 2025 opposite adjacent tan x° tan CD 450, AC 2025 x tan1 Solve for x. 450 2025 x 12.5 Use a calculator. The angle of elevation for the takeoff should be more than 12.5°. Lesson 7-5 Angles of Elevation and Depression 371 Robert Holmes/CORBIS ANGLES OF DEPRESSION An angle of depression B angle of depression is the angle between the line of sight when an observer looks downward, and the horizontal. A line of sight C D Standardized Example 2 Angle of Depression Test Practice Short-Response Test Item 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°. Find the length of the ramp to the nearest tenth foot. 10˚ 3.5 ft 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 Method 1 Method 2 10˚ A D A 3.5 ft 80˚ D 3.5 ft C 10˚ B The ground and the horizontal level with the back of the van are parallel. Therefore, mDABmABC since they are alternate interior angles. 3.5 AB sin 10° C The horizontal line from the back of the van and the segment from the ground to the back of the van are perpendicular. So, DAB and BAC are complementary angles. Therefore, mBAC 90 10 or 80. AB sin 10° 3.5 3.5 sin 10° AB B 10˚ 3.5 AB cos 80° AB cos 80° 3.5 AB 20.2 The ramp is about 20.2 feet long. 3.5 cos 80° AB AB 20.2 Angles of elevation or depression to two different objects can be used to find the distance between those objects. Study Tip 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. 372 Example 3 Indirect Measurement Olivia is 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. CDA and CDB are right triangles, and and CD110 85 or 195. The distance between the boats is AB or BD AD. Use the right triangles to find these two lengths. Chapter 7 Right Triangles and Trigonometry C E 33˚ 85 ft 57˚ 110 ft D A B Because C E and D B are horizontal lines, they are parallel. Thus, ECB CBD and ECA CAD because they are alternate interior angles. This means that mCBD 33 and mCAD 57. 195 DB tan 33° DB tan 33° 195 195 tan 33° opposite adjacent tan ; mCBD 33 Multiply each side by DB. DB Divide each side by tan 33°. DB 300.27 Use a calculator. 195 DA tan 57° DA tan 57° 195 opposite adjacent tan ; mCAD 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. Concept Check 1. OPEN ENDED Find a real-life example of an angle of depression. Draw a diagram and identify the angle of depression. P F 2. Explain why an angle of elevation is given that name. 3. Name the angles of depression and elevation in the figure. Guided Practice T B 4. 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.) 5. 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. 6. SALVAGE A salvage ship uses sonar to determine that the angle of depression to a wreck on the ocean floor is 13.25°. The depth chart shows that the ocean floor 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? Standardized Test Practice 7. SHORT RESPONSE From the top of a 150-foot high tower, an air traffic controller observes an airplane on the runway. To the nearest foot, how far from the base of the tower is the airplane? www.geometryonline.com/extra_examples/sol 13.25˚ 40 m 12˚ 150 ft Lesson 7-5 Angles of Elevation and Depression 373 Practice and Apply 8. 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? For Exercises See Examples 12, 14–18 9–11 8, 13, 19, 20 1 2 3 Extra Practice See page 768. 9. 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. 12˚ 36 yd 10. 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? 11. SLEDDING A sledding run is 300 yards long with a vertical drop of 27.6 yards. Find the angle of depression of the run. 12. RAILROADS The Monongahela Incline overlooks the city of Pittsburgh, Pennsylvania. Refer to the information at the left to determine the incline of the railway. 13. 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 CIVIL ENGINEERING For Exercises 14 and 15, use the following information. The percent grade of a highway is the ratio of the vertical rise or fall over a given horizontal distance. The ratio is expressed as a percent to the nearest whole number. Suppose a highway has a vertical rise of 140 feet for every 2000 feet of horizontal distance. 14. Calculate the percent grade of the highway. 15. Find the angle of elevation that the highway makes with the horizontal. Railroads The Monongahela Incline is 635 feet long with a vertical rise of 369.39 feet. It was built at a cost of $50,000 and opened on May 28, 1870. It is still used by commuters to and from Pittsburgh. Source: www.portauthority.org 16. 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 17 and 18, 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. 17. 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? 18. 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? 19. BIRDWATCHING 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? 374 Chapter 7 Right Triangles and Trigonometry R. Krubner/H. Armstrong Roberts 20. 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 MEDICINE For Exercises 21–23, 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°. 21. How far off the horizontal is the raised end of the treadmill at the beginning of the exam? 22. 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? 23. 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? Travel Ayers Rock is the largest monolith, a type of rock formation, in the world. It is approximately 3.6 kilometers long and 2 kilometers wide. The rock is believed to be the tip of a mountain, two thirds of which is underground. Source: www.atn.com.au 24. 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? 25. 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. orbit 29° 16° 3 mi n f Online Research Data Update Use the Internet to determine the angle of depression formed by someone aboard the international space station looking down to your community. Visit www.geometryonline.com/data_update to learn more. 26. CRITICAL THINKING 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. 27. WRITING IN MATH Answer the question that was posed at the beginning of the lesson. How do airline pilots use angles of elevation and depression? Include the following in your answer: • when pilots use angles of elevation or depression, and • the difference between angles of elevation and depression. www.geometryonline.com/self_check_quiz/sol Lesson 7-5 Angles of Elevation and Depression 375 John Mead/Science Photo Library/Photo Researchers Standardized Test Practice SOL/EOC Practice 28. The top of a signal tower is 120 meters above sea level. The angle of depression from the top of tower to a passing ship is 25°. How many meters from the foot of the tower is the ship? A 283.9 m B 257.3 m C 132.4 m 29. ALGEBRA A 56 m D y 28 25˚ 120 m x 16 1 2 7 4 If , then find x when y . 2 7 4 7 B C 1 2 3 D Maintain Your Skills Mixed Review Find the measure of each angle to the nearest tenth of a degree. (Lesson 7-4) 30. cos A 0.6717 31. sin B 0.5127 32. tan C 2.1758 33. cos D 0.3421 34. sin E 0.1455 35. tan F 0.3541 Find x and y. (Lesson 7-3) 36. 37. x 38. y 60˚ 14 12 20 y 30˚ 45˚ y x x 39. HOBBIES 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) 40. Copy and complete the flow proof. (Lesson 4-6) Given: 5 6 GS FR F 5 G 3 4 1 Prove: 4 3 X 6 2 R Proof: S 5 6 Given FR GS FXR GXS Given ? a. Vert. s are . Getting Ready for the Next Lesson b. ? ? c. d. e. ? f. ? ? PREREQUISITE SKILL Solve each proportion. (To review solving proportions, see Lesson 6-1.) x 35 41. 5 3 42. 12 24 43. x x 24 44. 12 48 45. x 5 46. 28 7 47. x 3 48. 6 13 42 x 376 Chapter 7 Right Triangles and Trigonometry x 18 45 8 17 15 x 36 40 15 26 The Law of Sines • Use the Law of Sines to solve triangles. • Solve problems by using the Law of Sines. are triangles used in radio astronomy? Vocabulary Study Tip 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. The Very Large Array (VLA), one of the world’s premier astronomical radio observatories, consists of 27 radio antennas in a Y-shaped configuration on the Plains of San Agustin in New Mexico. Astronomers use the VLA to make pictures from the radio waves emitted by astronomical objects. Construction of the antennas is supported by a variety of triangles, 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. 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 C b a sin A sin B sin C . a b c A Proof 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. sin A h b sin B h a b sin A h a sin B h Definition of sine C b A h D a B Cross products • Law of Sines • solving a triangle b sin A a sin B sin A sin B a b Substitution Divide each side by ab. The proof can be completed by using a similar technique with the other altitudes sin A a sin C c sin B b sin C c to show that and . Lesson 7-6 The Law of Sines 377 Example 1 Use the Law of Sines a. Find b. Round to the nearest tenth. B Use the Law of Sines to write a proportion. Study Tip Rounding If you round before the final answer, your results may differ from results in which rounding was not done until the final answer. sin A sin B a b sin 37° sin 68° 3 b b sin 37° 3 sin 68° 3 sin 68° sin 37° 68˚ Law of Sines 37˚ A mA 37, a 3, mB 68 3 C b Cross products b Divide each side by sin 37°. b 4.6 Use a calculator. b. Find mZ to the nearest degree in XYZ if y 17, z 14, and mY 92. Write a proportion relating Y, Z, y, and z. sin Y sin Z y z sin 92° sin Z 17 14 Law of Sines mY 92, y 17, z 14 14 sin 92° 17 sin Z Cross products 14 sin 92° sin Z 17 14 s in 92° sin1 Z 17 55° Z Divide each side by 17. Solve for Z. Use a calculator. So, mZ 55. 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. Study Tip Look Back To review the Angle Sum Theorem, see Lesson 4-2. Example 2 Solve Triangles a. Solve ABC if mA 33, mB 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 mC. C a b A c mA mB mC 180 Angle Sum Theorem 33 47 mC 180 mA 33, mB 47 80 mC 180 Add. mC 100 Subtract 80 from each side. sin B b Since we know mB and b, use proportions involving . To find a: To find c: sin B sin A b a sin 47° sin 33° 14 a sin B sin C b c sin 47° sin 100 ° 14 c a sin 47° 14 sin 33° 14 sin 33° sin 47° a a 10.4 Law of Sines Substitute. Cross products Divide each side by sin 47°. Use a calculator. Therefore, mC 100, a 10.4, and c 18.9. 378 Chapter 7 Right Triangles and Trigonometry c sin 47° 14 sin 100° 14 sin 100° sin 47° c c 18.9 B b. Solve ABC if mC 98, b 14, and c 20. Round angle measures to the nearest degree and side measures to the nearest tenth. Study Tip An Equivalent Proportion The Law of Sines may also be written as a b c . sin A sin B sin C You may wish to use this form when finding the length of a side. We know the measures of two sides and an angle opposite one of the sides. sin B sin C b c sin B sin 98° 14 20 Law of Sines mC 98, b 14, and c 20 20 sin B 14 sin 98° 14 sin 98° 20 14 sin 98° B sin–1 20 Cross products sin B Divide each side by 20. B 44° Solve for B. Use a calculator. mA mB mC 180 Angle Sum Theorem mA 44 98 180 mB 44 and mC 98 mA 142 180 mA 38 sin A sin C a c sin 38° sin 98° a 20 Add. Subtract 142 from each side. Law of Sines mA 38, mC 98, and c 20 20 sin 38° a sin 98° Cross products 20 sin 38° a sin 98° Divide each side by sin 98°. 12.4 a Use a calculator. Therefore, A 38°, B 44°, and a 12.4. USE THE LAW OF SINES TO SOLVE PROBLEMS The Law of Sines is very useful in solving direct and indirect measurement applications. Example 3 Indirect Measurement 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 of 30 feet long on the ground. Find the length of the telephone pole to the nearest tenth of a foot. Draw a diagram. S P GD Draw SD . Then find the mGDP and mGPD. 7° mGDP 90 7 or 83 pole mGPD 62 83 180 or mGPD 35 Since you know the measures of two angles of the triangle, mGDP and mGPD, and the length of a side opposite one of the angles (G D is opposite GPD) you can use the Law of Sines to find the length of the pole. www.geometryonline.com/extra_examples/sol G 62° 30-foot shadow D (continued on the next page) Lesson 7-6 The Law of Sines 379 PD GD sin DGP sin GPD PD 30 sin 62° sin 35° PD sin 35° 30 sin 62° 30 sin 62° sin 35° Law of Sines mDGP 62, mGPD 35, and GD 30 Cross products PD Divide each side by sin 35°. PD 46.2 Use a calculator. The telephone pole is about 46.2 feet long. 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) Concept Check 1. FIND THE ERROR Makayla and Felipe are trying to find d in DEF. Makayla d sin 59° 12 F Felipe 12 73˚ sin 59° sin 48° d 12 D Who is correct? Explain your reasoning. d 48˚ f E 2. 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. 3. Compare the two cases for the Law of Sines. Guided Practice Find each measure using the given measures of XYZ. Round angle measures to the nearest degree and side measures to the nearest tenth. 4. Find y if x 3, mX 37, and mY 68. 5. Find x if y 12.1, mX 57, and mZ 72. 6. Find mY if y 7, z 11, and mZ 37. 7. Find mZ if y 17, z 14, and mY 92. Solve each PQR described below. Round angle measures to the nearest degree and side measures to the nearest tenth. 8. mR 66, mQ 59, p 72 9. p 32, r 11, mP 105 10. mP 33, mR 58, q 22 11. p 28, q 22, mP 120 12. mP 50, mQ 65, p 12 13. q 17.2, r 9.8, mQ 110.7 14. Find the perimeter of parallelogram ABCD to the nearest tenth. A B 6 D 380 Chapter 7 Right Triangles and Trigonometry 32° 88° C Application 15. 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 from A to B as 62° and measures the angle from C to B as 55°. Find the distance from A to B. C A B Practice and Apply For Exercises See Examples 16–21 22–29 30–38 1 2 3 Extra Practice See page 768. Find each measure using the given measures of KLM. Round angle measures to the nearest degree and side measures to the nearest tenth. 16. If mL 45, mK 63, and 22, find k. 17. If k 3.2, mL 52, and mK 70, find . 18. If m 10.5, k 18.2, and mK 73, find mM. 19. If k 10, m 4.8, and mK 96, find mM. 20. If mL 88, mK 31, and m 5.4, find . 21. If mM 59, 8.3, and m 14.8, find mL. Solve each WXY described below. Round measures to the nearest tenth. 22. mY 71, y 7.4, mX 41 23. x 10.3, y 23.7, mY 96 24. mX 25, mW 52, y 15.6 25. mY 112, x 20, y 56 26. mW 38, mY 115, w 8.5 27. mW 36, mY 62, w 3.1 28. w 30, y 9.5, mW 107 29. x 16, w 21, mW 88 30. An isosceles triangle has a base of 46 centimeters and a vertex angle of 44°. Find the perimeter. 31. Find the perimeter of quadrilateral ABCD to the nearest tenth. A 12 B 28˚ 40˚ D C 32. 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. 33. AVIATION Two radar stations that are 20 miles apart located a plane at the same time. The first station indicated that the position of the plane made an angle of 43° with the line between the stations. The second station indicated that it made an angle of 48° with the same line. How far is each station from the plane? 34. SURVEYING Maria Lopez is a surveyor who must determine the distance across a section of the Rio Grande Gorge in New Mexico. Standing 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. Lesson 7-6 The Law of Sines 381 35. 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? 160 ft 85˚ 160 ft MIRRORS For Exercises 36 and 37, use the following information. Kayla, Jenna, and Paige live in a town nicknamed “Perpendicular City” because the planners and builders took great care to have all the streets oriented north-south or east-west. The three of them play a game where they signal each other using mirrors. Kayla and Jenna signal each other from a distance of 1433 meters. Jenna turns 27° to signal Paige. Kayla rotates 40° to signal Paige. 36. To the nearest tenth of a meter, how far apart are Kayla and Paige? 37. To the nearest tenth of a meter, how far apart are Jenna and Paige? Aviation From January 2001 to May 2001, Polly Vacher flew over 29,000 miles to become the first woman to fly the smallest single-engined aircraft around the world via Australia and the Pacific. Source: www.worldwings.org AVIATION For Exercises 38 and 39, use the following information. Keisha Jefferson is flying a small plane due west. To avoid the jet stream, she must change her course. She turns the plane 27° to the south and flies 60 miles. Then she makes a turn of 124° heads back to her original course. 27˚ 124˚ 38. How far must she fly after the second turn to return to the original course? 39. How many miles did she add to the flight by changing course? 40. CRITICAL THINKING Does the Law of Sines apply to the acute angles of a right triangle? Explain your answer. 41. WRITING IN MATH Answer the question that was posed at the beginning of the lesson. How are triangles used in radio astronomy? Include the following in your answer: • a description of what the VLA is and the purpose of the VLA, and • the purpose of the triangles in the VLA. SOL/EOC Practice Standardized Test Practice 42. SHORT RESPONSE In XYZ, if x 12 , mX 48, and mY 112, solve the triangle to the nearest tenth. 43. ALGEBRA The table below shows the customer ratings of five restaurants in the Metro City Guide to Restaurants. The rating scale is from 1, the worst, to 10, the best. Which of the five restaurants has the best average rating? Restaurant A C Food Decor Service Value Del Blanco’s 7 9 4 7 Aquavent 8 9 4 6 Le Circus 10 8 3 5 Sushi Mambo 7 7 5 6 Metropolis Grill 9 8 7 7 Metropolis Grill Aquavent 382 Chapter 7 Right Triangles and Trigonometry B D Le Circus Del Blanco’s Maintain Your Skills Mixed Review ARCHITECTURE For Exercises 44 and 45, use the following information. x 1 ft Mr. Martinez is an architect who designs houses so that the windows receive minimum sun in the summer and window maximum sun in the winter. For Columbus, Ohio, the angle of 6 ft angle of elevation of the sun at noon on the longest day elevation high of sun 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 7-5) 44. How long should the architect make the overhang so that the window gets no direct sunlight at noon on the longest day? 45. Using the overhang from Exercise 44, how much of the window will get direct sunlight at noon on the shortest day? J 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 7-4) 46. j 8, k 17, 15 47. j 20, k 29, 21 49. j 72, k 14, 72 48. j 12, k 24, 123 If K H is parallel to JI, find the measure of each angle. (Lesson 4-2) 50. 1 51. 2 52. 3 Getting Ready for the Next Lesson L K 54˚ 36˚ 120˚ 1 3 J 2 I H c2 a2 b2 2ab PREREQUISITE SKILL Evaluate for the given values of a, b, and c. (To review evaluating expressions, see page 736.) 53. a 7, b 8, c 10 56. a 16, b 4, c 13 54. a 4, b 9, c 6 57. a 3, b 10, c 9 P ractice Quiz 2 Find x to the nearest tenth. (Lesson 7-4) 1. 2. 16 K Lessons 7-4 through 7-6 x 3. 10 x˚ 55. a 5, b 8, c 10 58. a 5, b 7, c 11 9.7 4. 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. 17˚ 32 53˚ 5. Solve DEF. x D 12 E (Lesson 7-6) 82˚ 8 F (Lesson 7-5) www.geometryonline.com/self_check_quiz/sol Lesson 7-6 The Law of Sines 383 A Follow-Up of Lesson 7-6 The Ambiguous Case of the Law of Sines In Lesson 7-6, 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. Model and AC . Construct a Construct AB circle whose center is B so that it at two points. Then, intersects AC construct any radius BD . Using the rotate tool, move D to the other intersection point of circle B . and AC Step 5 Find the measures of B D, A B, and A. Note the measures of ABD, BDA, and AD in ABD. Use the rotate tool to move D so that it lies on one of the intersection points . In ABD, find the of circle B and AC D. measures of ABD, BDA, and A 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 1. Which measures are the same in both triangles? 2. Repeat the activity using different measures for A, BD , and A B . How do the results compare to the earlier results? Make a Conjecture 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 B D , A B , and A, to have no solution? Test your conjecture and explain. 384 Investigating Slope-Intercept Form 384 Chapter 7 Right Triangles and Trigonometry The Law of Cosines • Use the Law of Cosines to solve triangles. • Solve problems by using the Law of Cosines. are triangles used in building design? Vocabulary • Law of Cosines The Chicago Metropolitan Correctional Center is a 27-story triangular federal detention center. The cells are arranged around a lounge-like common area. The architect found that a triangular floor plan allowed for the maximum number of cells to be most efficiently centered around the lounge. 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. Law of Cosines Study Tip Side and Angle 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. 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. C b A a2 b2 c 2 2bc cos A a B c b 2 a2 c 2 2ac cos B c 2 a2 b 2 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 1 Two Sides and the Included Angle Find a if c 8, b 10, and mA 60. Use the Law of Cosines since the measures of two sides and the included are known. a2 b2 c2 2bc cos A a2 102 82 2(10)(8) cos 60° a2 164 160 cos 60° A 60˚ 10 8 Law of Cosines b 10, c 8, and mA 60 C a B Simplify. a 164 160 cos 60° Take the square root of each side. a 9.2 Use a calculator. Lesson 7-7 The Law of Cosines 385 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 2 Three Sides Q 23 18 Find mR. r2 q2 s2 2qs cos R R S 37 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 13 32 Divide each side by –1332. 1332 1164 R cos1 Solve for R. R 29.1° Use a calculator. 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. Example 3 Select a Strategy Study Tip 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. Solve KLM. Round angle measure to the nearest degree and side measure 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 Sines. We know the measures of two sides and the included angle. This is SAS, so use the Law of Cosines. k2 2 m2 2m cos K k2 182 142 L 14 K k 51˚ 18 M Law of Cosines 2(18)(14) cos 51° 18, m 14, and mK 51 2 2(18)(1 k 182 14 cos 51° 4) Take the square root of each side. k 14.2 Use a calculator. Next, we can find mL or mM. If we decide to find mL, 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 k sin L sin 51° 18 14.2 Law of Sines 18, k 14.2, and mK 51 14.2 sin L 18 sin 51° 18 sin 51° 14.2 18 sin 51° L sin–1 14.2 Cross products sin L L 80° 386 Chapter 7 Right Triangles and Trigonometry Divide each side by 14.2. Take the inverse sine of each side. Use a calculator. Use the Angle Sum Theorem to find mM. mK mL mM 180 Angle Sum Theorem 51 80 mM 180 mK 51 and mL 80 mM 49 Subtract 131 from each side. Therefore, k 14.2, mK 80, and mM 49. Example 4 Use Law of Cosines to Solve Problems REAL ESTATE Ms. Jenkins is buying some property that is shaped like quadrilateral ABCD. Find the perimeter of the property. Use the Pythagorean Theorem to find BD in ABD. (AB)2 (AD)2 (BD)2 1802 AB 180, AD 240 90,000 (BD)2 Simplify. 300 BD B Pythagorean Theorem (BD)2 2402 C 200 ft 60˚ 180 ft A 240 ft D Take the square root of each side. Next, use the Law of Cosines to find CD in BCD. (CD)2 (BC)2 (BD)2 2(BC)(BD) cos CBD (CD)2 2002 3002 2(200)(300) cos 60° (CD)2 130,000 120,000 cos 60° Law of Cosines BC 200, BD 300, mCBD 60 Simplify. CD 130,00 0 120,000 60° cos Take the square root of each side. CD 264.6 Use a calculator. The perimeter of the property is 180 200 264.6 240 or about 884.6 feet. Concept Check 1. 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. 2. Explain when you should use the Law of Sines or the Law of Cosines to solve a triangle. 3. 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. Guided Practice In BCD, given the following measures, find the measure of the missing side. 5. b 107, c 94, mD 105 4. c 2, d 5, mB 45 In RST, given the lengths of the sides, find the measure of the stated angle to the nearest degree. 7. r 2.2, s 1.3, t 1.6; mR 6. r 33, s 65, t 56; mS Solve each triangle using the given information. Round angle measure to the nearest degree and side measure to the nearest tenth. 8. XYZ: x 5, y 10, z 13 9. KLM: k 20, m 24, mL 47 www.geometryonline.com/extra_examples/sol Lesson 7-7 The Law of Cosines 387 Application 10. CRAFTS Jamie, age 25, is creating a logo for herself and two cousins, ages 10 and 5. She is using a quarter (25 cents), a dime (10 cents), and a nickel (5 cents) to represent their ages. She will hold the coins together by soldering a triangular piece of silver wire so that the three vertices of the triangle lie at the centers of the three circular coins. The diameter of the quarter is 24 millimeters, the diameter of the nickel is 22 millimeters, and the diameter of a dime is 10 millimeters. Find the measures of the three angles in the triangle. Q D N Practice and Apply For Exercises See Examples 11–14 15–18, 38 22–37, 39–41 1 2 3 Extra Practice See page 768. In TUV, given the following measures, find the measure of the missing side. 12. t 11, u 17, mV 78 11. t 9.1, v 8.3, mU 32 13. u 11, v 17, mT 105 14. v 11, u 17, mT 59 In EFG, given the lengths of the sides, find the measure of the stated angle to the nearest degree. 16. e 14, f 19, g 32; mE 15. e 9.1, f 8.3, g 16.7; mF 17. e 325, f 198, g 208; mF 18. e 21.9, f 18.9, g 10; mG Solve each triangle using the given information. Round angle measures to the nearest degree and side measures to the nearest tenth. Q G 19. 20. 21. C 8 40˚ F 10 11 g H P p 11 38˚ 10 18 M B D 15 22. ABC: mA 42, mC 77, c 6 23. ABC: a 10.3, b 9.5, mC 37 24. ABC: a 15, b 19, c 28 25. ABC: mA 53, mC 28, c 14.9 26. KITES Beth is building a kite like the one at the right. If AB is 5 feet long, BC is 8 feet long, 2 D is 7 feet long, find the measure of the and B 3 angle between the short sides and the angle between the long sides to the nearest degree. A B D C Solve each LMN described below. Round measures to the nearest tenth. 27. m 44, 54, mL 23 28. m 18, 24, n 30 29. m 19, n 28, mL 49 30. mM 46, mL 55, n 16 31. m 256, 423, n 288 32. mM 55, 6.3, n 6.7 33. mM 27, 5, n 10 34. n 17, m 20, 14 35. 14, n 21, mM 60 36. 14, m 15, n 16 37. mL 51, 40, n 35 38. 10, m 11, n 12 388 Chapter 7 Right Triangles and Trigonometry 39. In quadrilateral ABCD, AC 188, BD 214, and mBPC 70, and P is the midpoint of AC D . Find the perimeter of quadrilateral ABCD. B A B P D 40. In quadrilateral PQRS, PQ 721, QR 547, RS 593, PS 756, and mP 58. Find QS, mPQS, and mR. C Q P 41. BUILDINGS Refer to the information at the left. Find the measures of the angles of the triangular building to the nearest tenth. 42. 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? R S Carlos 40 ft 50 ft 24 ft 30 ft 22 ft Adam Buildings 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 43. PROOF Justify each statement for the derivation of the Law of Cosines. A D is an altitude of ABC. Given: A b c h Prove: c2 a2 b2 2ab cos C B Proof: ax x C a Statement a. c2 (a x)2 h2 b. c2 a2 2ax x2 h2 c. x2 h2 b2 d. c2 a2 2ax b2 x e. cos C Reasons a. ? b. ? c. ? d. ? e. ? f. b cos C x g. c2 a2 2a(b cos C) b2 h. c2 a2 b2 2ab cos C f. g. h. b ? ? ? 44. CRITICAL THINKING Graph A(–6, –8), B(10, –4), C(6, 8), and D(5, 11) on a coordinate plane. Find the measure of interior angle ABC and the measure of exterior angle DCA. 45. WRITING IN MATH Answer the question that was posed at the beginning of the lesson. How are triangles used in building construction? Include the following in your answer: • why the building was triangular instead of rectangular, and • why the Law of Sines could not be used to solve the triangle. www.geometryonline.com/self_check_quiz/sol Lesson 7-7 The Law of Cosines 389 Standardized Test Practice SOL/EOC Practice 46. For DEF, find d to the nearest tenth if e 12, f 15, and mD 75. A 18.9 B 16.6 C 15.4 D 9.8 47. ALGEBRA Ms. LaHue earns a monthly base salary of $1280 plus a commission of 12.5% of her total monthly sales. At the end of one month, Ms. LaHue earned $4455. What were her total sales for the month? A $3175 B $10,240 C $25,400 D $35,640 Maintain Your Skills Mixed Review Find each measure using the given measures from XYZ. Round angle measure to the nearest degree and side measure to the nearest tenth. (Lesson 7-6) 48. If y 4.7, mX 22, and mY 49, find x. 49. If y 10, x 14, and mX 50, find mY. 50. 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 7-5) For Exercises 51–54, determine whether A B D. (Lesson 6-4) C 51. AC 8.4, BD 6.3, DE 4.5, and CE 6 52. AC 7, BD 10.5, BE 22.5, and AE 15 53. AB 8, AE 9, CD 4, and CE 4 54. AB 5.4, BE 18, CD 3, and DE 10 B A C D E Use the figure at the right to write a paragraph proof. (Lesson 6-3) H 55. Given: JFM EFB LFM GFB E Prove: JFL EFG D E 56. Given: JM B A LM G B G J L F M B Prove: JL E G 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) 57. orthocenter 58. centroid 59. circumcenter Who is Behind This Geometry Idea Anyway? It’s time to complete your project. Use the information and data you have gathered about your research topic, two mathematicians, and a geometry problem to prepare a portfolio or Web page. Be sure to include illustrations and/or tables in the presentation. www.geometryonline.com/webquest 390 Chapter 7 Right Triangles and Trigonometry C A Follow-Up of Lesson 7-7 Trigonometric Identities In algebra, the equation 2(x 2) 2x 4 is called an identity because the equation is true for all values of x. There are equations involving trigonometric ratios that are true for all values of the angle measure. These are called trigonometric identities . y In the figure, P(x, y) is in Quadrant I. The Greek letter theta (pronounced THAY tuh) , represents the measure of the angle . Triangle POR is a right triangle. formed by the x-axis and OP Let r represent the length of the hypotenuse. Then the following are true. y sin r r csc y 1 sin y 1 y 1 r 1 Notice that y r y y tan x x cot y x cos r r sec x P (x, y) r O y r r y r y 1 1 or r x R x r csc . y 1 So, csc . This is known as one of the reciprocal identities . sin Activity Verify that cos2 sin2 1. The expression cos2 means (cos )2. To verify an identity, work on only one side of the equation and use what you know to show how that side is equivalent to the other side. cos2 sin2 1 y xr2 r2 1 y2 x2 1 2 r r2 x2 y2 1 r2 r2 2 1 r 11 Original equation Substitute. Simplify. Combine fractions with like denominators. Pythagorean Theorem: x2 y2 r2 Simplify. Since 1 1, cos2 sin2 1. Analyze 1. The identity cos2 sin2 1 is known as a Pythagorean identity . Why do you think the word Pythagorean is used to name this? 1 cos 1 tan 2. Find two more reciprocal identities involving and . Verify each identity. sin 3. tan cos cos sin 4. cot 5. tan2 1 sec2 6. cot2 1 csc2 Geometry Activity Trigonometric Identities 391 Vocabulary and Concept Check ambiguous case (p. 384) angle of depression (p. 372) angle of elevation (p. 371) cosine (p. 364) geometric mean (p. 342) Law of Cosines (p. 385) Law of Sines (p. 377) Pythagorean identity (p. 391) Pythagorean triple (p. 352) reciprocal identities (p. 391) sine (p. 364) solving a triangle (p. 378) tangent (p. 364) trigonometric identity (p. 391) trigonometric ratio (p. 364) trigonometry (p. 364) A complete list of postulates and theorems can be found on pages R1–R8. Exercises State whether each statement is true or false. If false, replace the underlined word or words to make a true sentence. 1. 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. 2. The tangent of A is the measure of the leg adjacent to A divided by the measure of the leg opposite A. 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 elevation is the angle between the line of sight and the horizontal when an observer looks upward. 5. The geometric mean between two numbers is the positive square root of their product. 6. In a 30°-60°-90° triangle, two of the sides will have the same length. 7. Looking at a city while flying in a plane is an example that uses angle of elevation. 7-1 Geometric Mean See pages 342–348. Concept Summary • The geometric mean of two numbers is the square root of their product. • You can use the geometric mean to find the altitude of a right triangle. Examples 1 Find the geometric mean between 10 and 30. x 10 30 x x2 300 x 300 or 103 Definition of geometric mean Cross products Take the square root of each side. 2 Find NG in TGR. T The measure of the altitude is the geometric mean between the measures of the two hypotenuse segments. TN GN GN RN 2 GN GN 4 8 (GN)2 8 or 22 GN 392 Chapter 7 Right Triangles and Trigonometry Definition of geometric mean 2 N 4 G R TN 2, RN 4 Cross products Take the square root of each side. www.geometryonline.com/vocabulary_review Chapter 7 Study Guide and Review Exercises Find the geometric mean between each pair of numbers. See Example 1 on page 342. 8. 4 and 16 9. 4 and 81 10. 20 and 35 12. In PQR, PS 8, and QS 14. Find RS. See Example 2 on page 344. 11. 18 and 44 P S Q R 7-2 The Pythagorean Theorem and Its Converse See pages 350–356. Example Concept Summary • The Pythagorean Theorem can be used to find the measures of the sides of a right triangle. • If the measures of the sides of a triangle form a Pythagorean triple, then the triangle is a right triangle. Find k. a2 (LK)2 (JL)2 a2 82 132 a2 64 169 a2 105 a 105 a 10.2 Exercises 13. 15 J Pythagorean Theorem LK 8 and JL 13 13 a Simplify. Subtract 64 from each side. K Take the square root of each side. L 8 Use a calculator. Find x. See Example 2 on page 351. 13 14. x 17 15. 5 17 x 13 x 21 20 7-3 Special Right Triangles See pages 357–363. Concept Summary • The measure of the hypotenuse of a 45°-45°-90° triangle is 2 times the length of the legs of the triangle. The measures of the sides are x, x, and x2. • In a 30°-60°-90° triangle, the measures of the sides are x, x3, and 2x. Examples 1 Find x. The measure of the shorter leg XZ of XYZ is half the X 60˚ 1 2 measure of the hypotenuse XY . Therefore, XZ (26) or 13. The measure of the longer leg is 3 times the measure of the shorter leg. So, x 133. Z 26 x Y Chapter 7 Study Guide and Review 393 Chapter 7 Study Guide and Review 2 Find x. Q The measure of the hypotenuse of a 45°-45°-90° triangle 4 is 2 times the length of a leg of the triangle. x2 4 45˚ P 4 x 2 4 2 x or 22 2 2 Exercises x R Find x and y. See Examples 1 and 3 on pages 358 and 359. 16. 17. y 45˚ x 9 18. x 13 30˚ x y 45˚ 60˚ y x 6 y 60˚ For Exercises 19 and 20, use the figure at the right. See Example 3 on page 359. x 19. If y 18, find z and a. 20. If x 14, find a, z, b, and y. b z 30˚ a 7-4 Trigonometry See pages 364–370. Example Concept Summary • Trigonometric ratios can be used to find measures in right triangles. Find sin A, cos A, and tan A. Express each ratio as a fraction and as a decimal. B 13 A opposite leg hypotenuse BC AB 5 or about 0.38 13 sin A adjacent leg hypotenuse AC AB 12 or about 0.92 13 cos A 5 C 12 opposite leg adjacent leg BC AC 5 or about 0.42 12 tan A Exercises 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. See Example 1 on page 365. 21. a 9, b 12, c 15 22. a 7, b 24, c 25 G F Find the measure of each angle to the nearest tenth of a degree. See Example 4 on pages 366 and 367. 23. sin P 0.4522 394 Chapter 7 Right Triangles and Trigonometry 24. cos Q 0.1673 25. tan R 0.9324 c a b H Chapter 7 Study Guide and Review 7-5 Angles of Elevation and Depression See pages 371–376. Example Concept Summary • Trigonometry can be used to solve problems related to angles of elevation and depression. A store has a ramp near its front entrance. The ramp measures 12 feet, and has a height of 3 feet. What is the angle of elevation? Make a drawing. B 12 Let x represent mBAC. BC AB 3 sin x° 12 opposite hypotenuse sin x° sin x A 3 x˚ C BC 3 and AB 12 x sin-1 Find the inverse. 3 12 x 14.5 Use a calculator. The angle of elevation for the ramp is about 14.5°. Exercises Determine the angles of elevation or depression in each situation. See Examples 1 and 2 on pages 371 and 372. 26. An airplane must clear a 60-foot pole at the end of a runway 500 yards long. 27. An escalator descends 100 feet for each horizontal distance of 240 feet. 28. A hot-air balloon descends 50 feet for every 1000 feet traveled horizontally. 29. DAYLIGHT At a certain time of the day, the angle of elevation of the sun is 44°. Find the length of a shadow cast by a building that is 30 yards high. 30. RAILROADS A railroad track rises 30 feet for every 400 feet of track. What is the measure of the angle of elevation of the track? 7-6 The Law of Sines See pages 377–383. Example Concept Summary • To find the measures of a triangle by using the Law of Sines, you must either know the measures of two angles and any side (AAS or ASA), or two sides and an angle opposite one of these sides (SSA) of the triangle. • To solve a triangle means to find the measures of all sides and angles. Solve XYZ if mX 32, mY 61, and y 15. Round angle measures to the nearest degree and side measures to the nearest tenth. Find the measure of Z. Y mX mY mZ 180 Angle Sum Theorem 61˚ z x 32 61 mZ 180 mX 32 and mY 61 93 mZ 180 Add. mZ 87 Subtract 93 from each side. X 32˚ y Z (continued on the next page) Chapter 7 Study Guide and Review 395 • Extra Practice, see pages 766-768. • Mixed Problem Solving, see page 788. Since we know mY and y, use proportions involving sin Y and y. To find x: To find z: sin Y sin X y x sin 61° sin 32° 15 x Substitute. x sin 61° 15 sin 32° Cross products 15 sin 32° x sin 61° x 9.1 sin Y sin Z y z sin 61° sin 87° 15 z Law of Sines z sin 61° 15 sin 87° 15 sin 87° sin 61° z Divide. z 17.1 Use a calculator. Exercises Find each measure using the given measures of FGH. Round angle measures to the nearest degree and side measures to the nearest tenth. See Example 1 on page 378. 31. Find f if g 16, mG 48, and mF 82. 32. Find mH if h 10.5, g 13, and mG 65. Solve each ABC described below. Round angle measures to the nearest degree and side measures to the nearest tenth. See Example 2 on pages 378 and 379. 33. a 15, b 11, mA 64 34. c 12, mC 67, mA 55 35. mA 29, a 4.8, b 8.7 36. mA 29, mB 64, b 18.5 7-7 The Law of Cosines See pages 385–390. Example Concept Summary • The Law of Cosines can be used to solve triangles when you know the measures of two sides and the included angle (SAS) or the measures of the three sides (SSS). Find a if b 23, c 19, and mA 54. Since the measures of two sides and the included angle are known, use the Law of Cosines. Law of Cosines a2 b c2 2bc cos A a2 232 192 2(23)(19) cos 54° b 23, c 19, and mA 54 a 232 192 2(23)(1 cos 54° 9) Take the square root of each side. a 19.4 Use a calculator. B 19 A a 54˚ 23 Exercises In XYZ, given the following measures, find the measure of the missing side. See Example 1 on page 385. 37. x 7.6, y 5.4, mZ 51 38. x 21, mY 73, z 16 Solve each triangle using the given information. Round angle measure to the nearest degree and side measure to the nearest tenth. See Example 3 on pages 386 and 387. 39. c 18, b 13, mA 64 396 Chapter 7 Right Triangles and Trigonometry 40. b 5.2, mC 53, c 6.7 C Vocabulary and Concepts 1. State the Law of Cosines for ABC used to find mC. 2. Determine whether the geometric mean of two perfect squares will always be rational. Explain. 3. Give an example of side measures of a 30°-60°-90° triangle. Skills and Applications Find the geometric mean between each pair of numbers. 5. 6 and 24 4. 7 and 63 6. 10 and 50 Find the missing measures. 7. 9. 6 8. 7 x 9 x x 5 12 10. 11. y 12. y 19 x 9 13 16 x y˚ 60˚ 12 45˚ 14. tan A x B Use the figure to find each trigonometric ratio. Express answers as a fraction. 13. cos B 8 21 15. sin A A Find each measure using the given measures from FGH. Round to the nearest tenth. 16. Find g if mF 59, f 13, and mG 71. 16 15 C 17. Find mH if mF 52, f 10, and h 12.5. 18. Find f if g 15, h 13, and mF 48. 19. Find h if f 13.7, g 16.8, and mH 71. Solve each triangle. Round each angle measure to the nearest degree and each side measure to the nearest tenth. 20. a 15, b 17, mC 45 21. a 12.2, b 10.9, mB 48 22. a 19, b 23.2, c 21 23. 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? 24. 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? 25. STANDARDIZED TEST PRACTICE Find tan X. SOL/EOC Practice A 5 12 B 12 13 C 17 12 www.geometryonline.com/chapter_test/sol X D 12 5 13 5 Chapter 7 Practice Test 397 SOL/EOC Practice Part 1 Multiple Choice Record your answers on the answer sheet provided by your teacher or on a sheet of paper. 4. In ABC, CD is an altitude and mACB 90°. If AD 12 and BD 3, find AC to the nearest tenth. (Lesson 7-1) C 1. If 4 and 3 are supplementary, which reason could you use as the first step in proving that 1 and 2 are supplementary? (Lesson 2-7) a b 1 2 4 3 A A 12 6.5 9.0 B D 3 B 13.4 C D 15.0 5. What is the length of R T ? (Lesson 7-3) R A Definition of similar angles B Definition of perpendicular lines C Definition of a vertical angle D Division Property 2. In ABC, BD is a median. If AD 3x 5 and CD 5x 1, find AC. (Lesson 5-1) C D 5 cm 135˚ T A 5 cm B 52 cm C 53 cm D 10 cm 6. An earthquake damaged a communication tower. As a result, the top of the tower broke off at a point 60 feet above the base. If the fallen portion of the tower made a 36° angle with the ground, what was the approximate height of the original tower? (Lesson 7-4) B A S A 3 B 11 C 14 D 28 60 ft 36° 3. If pentagons ABCDE and PQRST are similar, find SR. (Lesson 6-2) A 11 E D 2 3 C B 95 ft C 102 ft D 162 ft 6 T 8 35 ft P B 14 A Q S R 4 A 14 B C 3 D 4 11 5 1 6 398 Chapter 7 Right Triangles and Trigonometry 7. Miraku drew a map showing Regina’s house, Steve’s apartment, and Trina’s workplace. The three locations formed RST, where mR 34, r 14, and s 21. Which could be mS? (Lesson 7-6) A 15 B 22 C 57 D 84 Preparing for Standardized Tests For test-taking strategies and more practice, see pages 795– 810. Part 2 Short Response/Grid In Record your answers on the answer sheet provided by your teacher or on a sheet of paper. 8. Find mABC if mCDA = 61. A 69˚ (Lesson 1-6) B x˚ Test-Taking Tip Questions 6, 7, and 12 If a standardized test question involves trigonometric functions, draw a diagram that represents the problem and use a calculator (if allowed) or the table of trigonometric relationships provided with the test to help you find the answer. 55˚ C D For Questions 9 and 10, refer to the graph. 80 °F 12. Dee is parasailing at the ocean. The angle of depression from her line of sight to the boat is 41°. If the cable attaching Dee to the boat is 500 feet long, how many feet is Dee above the water? (Lesson 7-5) 60 (10, 50) 40 (0, 32) 41˚ 20 500 ft O 10 20 30 °C 9. At the International Science Fair, a Canadian student recorded temperatures in degrees Celsius. A student from the United States recorded the same temperatures in degrees Fahrenheit. They used their data to plot a graph of Celsius versus Fahrenheit. What is the slope of their graph? (Lesson 3-3) 10. Students used the equation of the line for the temperature graph of Celsius versus Fahrenheit to convert degrees Celsius to degrees Fahrenheit. If the line goes through points (0, 32) and (10, 50), what equation can the students use to convert degrees Celsius to degrees Fahrenheit? (Lesson 3-4) Part 3 Extended Response Record your answers on a sheet of paper. Show your work. 13. Toby, Rani, and Sasha are practicing for a double Dutch rope-jumping tournament. Toby and Rani are standing at points T and R and are turning the ropes. Sasha is standing at S, equidistant from both Toby and Rani. Sasha will jump into the middle of the turning rope to point X. Prove that when Sasha jumps into the rope, she will be at the midpoint between Toby and Rani. 11. TUV and XYZ are similar. Calculate the YZ ratio . (Lesson 6-3) (Lessons 4-5 and 4-6) UV S T X 10 V 6 U Z Y www.geometryonline.com/standardized_test/sol T X R Chapter 7 Standardized Test Practice 399

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