8-1 The Pythagorean Theorem and Its Converse Vocabulary Review 1. Write the square and the positive square root of each number. Number Square Positive Square Root 9 81 3 1 4 1 16 1 2 Vocabulary Builder leg po ten us e Related Word: hypotenuse Definition: In a right triangle, the sides that form the right angle are the legs. leg Main Idea: The legs of a right triangle are perpendicular. The hypotenuse is the side opposite the right angle. Use Your Vocabulary 2. Underline the correct word to complete the sentence. The hypotenuse is the longest / shortest side in a right triangle. Write T for true or F for false. F 3. The hypotenuse of a right triangle can be any one of the three sides. T 4. One leg of the triangle at the right has length 9 cm. T 5. The hypotenuse of the triangle at the right has length 15 cm. Chapter 8 202 12 cm 15 cm 9 cm Copyright © by Pearson Education, Inc. or its affiliates. All Rights Reserved. hy leg (noun) leg Theorems 8-1 and 8-2 Pythagorean Theorem and Its Converse Pythagorean Theorem If a triangle is a right triangle, then the sum of the squares of the lengths of the legs is equal to the square of the length of the hypotenuse. B c If nABC is a right triangle, then a2 1 b2 5 c2 . A a C b Converse of the Pythagorean Theorem If the sum of the squares of the lengths of two sides of a triangle is equal to the square of the length of the third side, then the triangle is a right triangle. If a2 1 b2 5 c2 , then nABC is a right triangle. 6. Circle the equation that shows the correct relationship among the lengths of the legs and the hypotenuse of a right triangle. 132 1 52 5 122 52 1 122 5 132 122 1 132 5 52 Underline the correct words to complete each sentence. 7. A triangle with side lengths 3, 4, and 5 is / is not a right triangle because 32 1 42 is equal / not equal to 52 . 8. A triangle with side lengths 4, 5, and 6 is / is not a right triangle because 42 1 52 is equal / not equal to 62 . Copyright © by Pearson Education, Inc. or its affiliates. All Rights Reserved. Problem 1 Finding the Length of the Hypotenuse Got It? The legs of a right triangle have lengths 10 and 24. What is the length of the hypotenuse? 9. Label the triangle at the right. 10. Use the justifications below to find the length of the hypotenuse. a2 1 b2 5 c2 2 10 Pythagorean Theorem c 24 2 10 1 24 5 c2 Substitute for a and b. 100 1 576 5 c2 Simplify. 676 5 c2 Add. 26 5 c Take the positive square root. 11. The length of the hypotenuse is 26 . 12. One Pythagorean triple is 5, 12, and 13. If you multiply each number by 2, what numbers result? How do the numbers that result compare to the lengths of the sides of the triangle in Exercises 9–11? 10, 24, 26. Answers may vary. Sample: The numbers are the same _______________________________________________________________________ as the lengths of the sides of the triangle in Exercises 9–11. _______________________________________________________________________ 203 Lesson 8-1 Problem 3 Finding Distance Got It? The size of a computer monitor is the length of its diagonal. You 19 in. want to buy a 19-in. monitor that has a height of 11 in. What is the width of the monitor? Round to the nearest tenth of an inch. 13. Label the diagram of the computer monitor at the right. 11 in. b in. 14. The equation is solved below. Write a justification for each step. a2 1 b2 5 c2 Pythagorean Theorem 112 1 b2 5 192 Substitute. 121 1 b2 5 361 Simplify. 121 2 121 1 b2 5 361 2 121 b2 5 240 Subtract 121 from each side. Simplify. b 5 "240 Take the positive square root. b < 15.49193338 Use a calculator. Problem 4 Identifying a Right Triangle Got It? A triangle has side lengths 16, 48, and 50. Is the triangle a right triangle? Explain. 16. Circle the equation you will use to determine whether the triangle is a right triangle. 162 1 482 0 502 162 1 502 0 482 482 1 502 0 162 17. Simplify your equation from Exercise 16. 162 1 482 0 502 256 1 2304 0 2500 2560 u 2500 18. Underline the correct words to complete the sentence. The equation is true / false , so the triangle is / is not a right triangle. A Pythagorean triple is a set of nonzero whole numbers a, b, and c that satisfy the equation a 2 1 b 2 5 c 2 . If you multiply each number in a Pythagorean triple by the same whole number, the three numbers that result also form a Pythagorean triple. Chapter 8 204 Copyright © by Pearson Education, Inc. or its affiliates. All Rights Reserved. 15. To the nearest tenth of an inch, the width of the monitor is 15.5 in. Theorems 8-3 and 8-4 Pythagorean Inequality Theorems Theorem 8-3 If the square of the length of the longest side of a triangle is greater than the sum of the squares of the lengths of the other two sides, then the triangle is obtuse. Theorem 8-4 If the square of the length of the longest side of a triangle is less than the sum of the squares of the lengths of the other two sides, then the triangle is acute. Use the figures at the right. Complete each sentence with acute or obtuse. 19. In nABC, c 2 . a 2 1 b 2 , so nABC is 9. A a C b obtuse S t 20. In nRST, s 2 , r 2 1 t 2 , so nRST is 9. B c acute R r T s Lesson Check • Do you UNDERSTAND? Error Analysis A triangle has side lengths 16, 34, and 30. Your friend says it is not a right triangle. Look at your friend’s work and describe the error. 21. Underline the length that your friend used as the longest side. Circle the length of the longest side of the triangle. 16 30 162 + 342 =? 302 256 + 1156 =? 900 1412 ≠ 900 34 Copyright © by Pearson Education, Inc. or its affiliates. All Rights Reserved. 22. Write the comparison that your friend should have used to determine whether the triangle is a right triangle. 162 1 302 0 342 23. Describe the error in your friend’s work. Answers may vary. Sample: My friend used the wrong length for c in the comparison. The comparison _______________________________________________________________________ should be 162 1 302 0 342 . _______________________________________________________________________ Math Success Check off the vocabulary words that you understand. hypotenuse leg Pythagorean Theorem Pythagorean triple Rate how well you can use the Pythagorean Theorem and its converse. Need to review 0 2 4 6 8 Now I get it! 10 205 Lesson 8-1 Special Right Triangles 8-2 Vocabulary Review 1. Circle the segment that is a diagonal of square ABCD. AB AC AD BC D C A B CD 2. Underline the correct word to complete the sentence. A diagonal is a line segment that joins two sides / vertices of a polygon. Vocabulary Builder complement (noun) KAHM pluh munt Other Word Form: complementary (adjective) Nonexample: Two angles whose measures sum to 180 are supplementary. Use Your Vocabulary Complete each statement with the word complement or complementary. 3. If m/A 5 40 and m/B 5 50, the angles are 9. complementary 4. If m/A 5 30 and m/B 5 60, /B is the 9 of /A. complement 5. /P and /Q are 9 because the sum of their measures is 90. complementary Complete. 6. If /R has a measure of 35, then the complement of /R has a measure of 55 . 7. If /X has a measure of 22, then the complement of /X has a measure of 68 . 8. If /C has a measure of 65, then the complement of /C has a measure of 25 . 9. Circle the complementary angles. 60í Chapter 8 40í 50í 206 120í Copyright © by Pearson Education, Inc. or its affiliates. All Rights Reserved. Math Usage: When the measures of two angles have a sum of 90, each angle is a complement of the other. Theorem 8-5 45°-45°-90° Triangle Theorem In a 458-458-908 triangle, both legs are congruent and the length of the hypotenuse is "2 times the length of a leg. 45 s 62 Complete each statement for a 458 2458 2908 triangle. 10. hypotenuse 5 "2 ? leg 45 s s 11. If leg 5 10, then hypotenuse 5 "2 ? 10 . Problem 1 Finding the Length of the Hypotenuse Got It? What is the length of the hypotenuse of a 458-458-908 triangle with leg length 5!3 ? 12. Use the justifications to find the length of the hypotenuse. hypotenuse 5 "2 ? leg 458-458-908 Triangle Theorem 5 "2 ? 5 "3 Substitute. 5 5 "2 ? "3 Commutative Property of Multiplication. 5 5 "6 Simplify. Copyright © by Pearson Education, Inc. or its affiliates. All Rights Reserved. Problem 2 Finding the Length of a Leg Got It? The length of the hypotenuse of a 458-458-908 triangle is 10. What is the length of one leg? 13. Will the length of the leg be greater than or less than 10? Explain. Less than. Explanations may vary. Sample: The hypotenuse is the longest side. __________________________________________________________________________________ 14. Use the justifications to find the length of one leg. hypotenuse 5 "2 ? leg 10 5 "2 ? leg 10 "2 5 leg 5 leg 5 leg 5 "2 "2 ? leg 10 "2 ? 10 "2 2 leg 5 5 "2 Substitute. Divide each side by "2 . Simplify. "2 10 458-458-908 Triangle Theorem "2 "2 Multiply by a form of 1 to rationalize the denominator. Simplify. Divide by 2. 207 Lesson 8-2 Problem 3 Finding Distance Got It? You plan to build a path along one diagonal of a 100 ft-by100 ft square garden. To the nearest foot, how long will the path be? 15. Use the words path, height, and width to complete the diagram. height 16. Write L for leg or H for hypotenuse to identify each part of the right triangle in the diagram. H path L height path L width width 17. Substitute for hypotenuse and leg. Let h 5 the length of the hypotenuse. hypotenuse 5 "2 ? leg h 5 "2 ? 100 18. Solve the equation. Use a calculator to find the length of the path. h 5 !2 ? 100 h N 141.4213562 19. To the nearest foot, the length of the path will be 141 feet. Theorem 8-6 30°-60°-90° Triangle Theorem In a 308-608-908 triangle, the length of the hypotenuse is twice the length of the shorter leg. The length of the longer leg is "3 times the length of the shorter leg. 2s 30 s V3 20. hypotenuse 5 2 60 s ? shorter leg 21. longer leg 5 "3 ? shorter leg Problem 4 Using the Length of One Side Got It? What is the value of f in simplest radical form? 5œ3 3 22. Complete the reasoning model below. Write f is the length of the hypotenuse. I can write an hypotenuse â 2 Ƃ shorter leg fâ 2 Ƃ Now I can solve for f. Chapter 8 fâ 208 10ƃ3 3 60˚ 30˚ f Think equation relating the hypotenuse and the 5ƃ3 shorter leg of the 30í-60í-90ítriangle. 3 5 5ƃ3 3 Copyright © by Pearson Education, Inc. or its affiliates. All Rights Reserved. Complete each statement for a 308-608-908 triangle. Problem 5 Applying the 30°-60°-90° Triangle Theorem Got It? Jewelry Making An artisan makes pendants in the shape of equilateral triangles. Suppose the sides of a pendant are 18 mm long. What is the height of the pendant to the nearest tenth of a millimeter? 18 mm 18 mm 23. Circle the formula you can use to find the height of the pendant. hypotenuse 5 2 ? shorter leg longer leg 5 !3 ? shorter leg 18 mm 24. Find the height of the pendant. longer leg 5 "3 ? shorter leg 5 "3 ? 9 N 15.58845727 25. To the nearest tenth of a millimeter, the height of the pendant is 15.6 mm. Lesson Check • Do you UNDERSTAND? Reasoning A test question asks you to find two side lengths of a 45°-45°-90° triangle. You know that the length of one leg is 6, but you forgot the special formula for 45°-45°-90° triangles. Explain how you can still determine the other side lengths. What are the other side lengths? 26. Underline the correct word(s) to complete the sentence. In a 45°-45°-90° triangle, Copyright © by Pearson Education, Inc. or its affiliates. All Rights Reserved. the lengths of the legs are different / the same . 27. Use the Pythagorean Theorem to find the length of the longest side. longest side: c2 5 62 1 62 c2 5 36 1 36 c2 5 72 c 5 "72 5 6 "2 28. The other two side lengths are 6 and 6 "2 . Math Success Check off the vocabulary words that you understand. leg hypotenuse right triangle Pythagorean Theorem Rate how well you can use the properties of special right triangles. Need to review 0 2 4 6 8 Now I get it! 10 209 Lesson 8-2 Trigonometry 8-3 Vocabulary Review The Venn diagram at the right shows the relationship between similar and congruent figures. Write T for true or F for false. F T T 1. All similar figures are congruent figures. Similar Figures Congruent Figures 2. All congruent figures are similar figures. 3. Some similar figures are congruent figures. 4. Circle the postulate or theorem you can use to verify that the triangles at the right are similar. AA , Postulate SAS , Theorem SSS , Theorem ratio (noun) RAY shee oh Related Words: rate, rational Definition: A ratio is the comparison of two quantities by division. 6 Example: If there are 6 triangles and 5 squares, the ratio of triangles to squares is 5 5 and the ratio of squares to triangles is 6 . Use Your Vocabulary Use the triangle at the right for Exercises 5–7. 5 13 5. Circle the ratio of the length of the longer leg to the length of the shorter leg. 5 13 5 12 12 13 13 12 12 5 13 5 6. Circle the ratio of the length of the shorter leg to the length of the hypotenuse. 5 13 5 12 12 13 13 12 12 5 13 5 7. Circle the ratio of the length of the longer leg to the length of the hypotenuse. 5 13 Chapter 8 5 12 12 13 13 12 12 5 210 13 5 12 Copyright © by Pearson Education, Inc. or its affiliates. All Rights Reserved. Vocabulary Builder Key Concept The Trigonometric Ratios sine of /A 5 B length of leg opposite/A a 5 length of hypotenuse c b length of leg adjacent to/A cosine of /A 5 5 c length of hypotenuse length of leg opposite/A tangent of /A 5 5 length of leg adjacent to/A a c A a C b b Draw a line from each trigonometric ratio in Column A to its corresponding ratio in Column B. Column A Column B a c b a b c 8. sin B 9. cos B 10. tan B 11. Reasoning Suppose nABC is a right isosceles triangle. What would the tangent of /B equal? Explain. Explanations may vary. Sample: 1. The legs would be congruent, so ba would equal 1. _______________________________________________________________________ Copyright © by Pearson Education, Inc. or its affiliates. All Rights Reserved. Problem 1 Writing Trigonometric Ratios G Got It? What are the sine, cosine, and tangent ratios for lG? 17 12. Circle the measure of the leg opposite /G. 8 15 17 T 15 8 R 13. Circle the measure of the hypotenuse. 8 15 17 14. Circle the measure of the leg adjacent to /G. 8 15 17 15. Write each trigonometric ratio. sin G 5 15 opposite 5 hypotenuse 17 cos G 5 8 adjacent 5 hypotenuse 17 tan G 5 15 opposite 5 adjacent 8 211 Lesson 8-3 Problem 2 Using a Trigonometric Ratio to Find Distance Got It? Find the value of w to the nearest tenth. 54 Below is one student’s solution. 17 w w 17 cos 54î(17) â w cos 54î â 9.992349289 Ƽ w 10 Ƽ w 16. Circle the trigonometric ratio that uses sides w and 17. sin 548 cos 548 tan 548 17. What error did the student make? w Answers may vary. Sample: The student wrote cos 54° 5 17 rather _______________________________________________________________________ w than sin 54° 5 17 . _______________________________________________________________________ 18. Find the value of w correctly. 19. The value of w to the nearest tenth is 13.8 . Problem 3 Using Inverses Got It? Use the figure below. What is mlY to the nearest degree? P 100 T 41 Y 20. Circle the lengths that you know. hypotenuse side adjacent to /Y side opposite /Y 21. Cross out the ratios that you will NOT use to find m/Y . sine cosine tangent 22. Underline the correct word to complete the statement. If you know the sine, cosine, or tangent ratio of an angle, you can use the inverse / ratio to find the measure of the angle. Chapter 8 212 Copyright © by Pearson Education, Inc. or its affiliates. All Rights Reserved. w sin 54° 5 17 sin 54°(17) 5 w 13.7532889 N w 13.8 N w 23. Follow the steps to find m/Y . Write the ratio. 1 tan Y â 100 41 Use the inverse. 2 3 Y â tanź1 ( ) 100 41 Use a calculator. YƼ 67.70637084 24. To the nearest degree, m/Y < 68° . Lesson Check • Do you UNDERSTAND? Error Analysis A student states that sin A S sin X because the lengths of the sides of kABC are greater than the lengths of the sides of kXYZ. What is the student’s error? Explain. Underline the correct word(s) to complete each sentence. Copyright © by Pearson Education, Inc. or its affiliates. All Rights Reserved. 25. nABC and nXYZ are / are not similar. B Y Z 35 X C 35 A 26. /A and /X are / are not congruent, so sin 358 is / is not equal to sin 358. 27. What is the student’s error? Explain. Answers may vary. Sample: The student did not look at the _________________________________________________________________ measures of lA and lX. Congruent angles have equal sine ratios. _________________________________________________________________ Math Success Check off the vocabulary words that you understand. trigonometric ratios sine cosine tangent Rate how well you can use trigonometric ratios. Need to review 0 2 4 6 8 Now I get it! 10 213 Lesson 8-3 8-4 Angles of Elevation and Depression Vocabulary Review Underline the correct word(s) or number to complete each sentence. 1. The measure of a right angle is greater / less than the measure of an acute angle and greater / less than the measure of an obtuse angle. 2. A right angle has a measure of 45 / 90 /180 . 3. Lines that intersect to form four right angles are parallel / perpendicular lines. 4. Circle the right angle(s) in the figure. /ACB /ADB /BAC /BAD /CBA /DBA A D B C elevation (noun) el uh VAY shun Related Word: depression Definition: The elevation of an object is its height above a given level, such as eye level or sea level. Math Usage: Angles of elevation and depression are acute angles of right triangles formed by a horizontal distance and a vertical height. Use Your Vocabulary Complete each statement with the correct word from the list below. Use each word only once. elevate elevated elevation 5. John 9 his feet on a footstool. elevated 6. The 9 of Mt McKinley is 20,320 ft. elevation 7. You 9 an object by raising it to a higher position. elevate Chapter 8 214 Copyright © by Pearson Education, Inc. or its affiliates. All Rights Reserved. Vocabulary Builder Problem 1 Identifying Angles of Elevation and Depression Got It? What is a description of l2 as it relates to the situation shown? Write T for true or F for false. T 8. /2 is above the horizontal line. T 9. /2 is the angle of elevation from the person in the hot-air balloon to the bird. F 10. /2 is the angle of depression from the person in the hot-air balloon to the bird. F 11. /2 is the angle of elevation from the top of the mountain to the person in the hot-air balloon. 12. Describe /2 as it relates to the situation shown. Copyright © by Pearson Education, Inc. or its affiliates. All Rights Reserved. Answers may vary. Sample: l2 is the angle of elevation from the _______________________________________________________________________ person in the hot-air balloon to the bird. _______________________________________________________________________ Problem 2 Using the Angle of Elevation Got It? You sight a rock climber on a cliff at a 32° angle of elevation. Your eye level is 6 ft above the ground and you are 1000 feet from the base of the cliff. What is the approximate height of the rock climber from the ground? 13. Use the information in the problem to complete the problem-solving model below. Know Angle of elevation is 32 Need Height of climber from the ground 8. Distance to the cliff is 6 32 1000 ft Eye level Plan Find the length of the leg opposite 328 by using tan 32 8 . Then add 1000 ft. Eye level is Climber 6 ft. ft above the ground. 215 Lesson 8-4 14. Explain why you use tan 328 and not sin 328 or cos 328. Answers may vary. Sample: The sine ratio involves two unknowns. The cosine ratio involves the _______________________________________________________________________ hypotenuse and 1000, but I do not want to know the hypotenuse. The _______________________________________________________________________ ratio that uses the unknown height and 1000 is the tangent ratio. _______________________________________________________________________ 15. The problem is solved below. Use one of the reasons from the list at the right to justify each step. d tan 328 5 1000 Write the equation. (tan 328) 1000 5 d Solve for d. Use a calculator. Write the equation. Solve for d. d < 624.8693519 Use a calculator. 16. The height from your eye level to the climber is about 625 ft. 17. The height of the rock climber from the ground is about 631 ft. Problem 3 Using the Angle of Depression Got It? An airplane pilot sights a life raft at a 26° angle of depression. The airplane’s altitude is 3 km. What is the airplane’s horizontal distance d from the raft? 26º altitude 3 km Not to scale Angle of elevation 26º d Angle of depression horizontal distance Raft 19. Circle the equation you could use to find the horizontal distance d. 3 sin 268 5 d 3 cos 268 5 d 20. Solve your equation from Exercise 19. tan 268 5 d3 d 5 tan3268 d 5 6.150911525 21. To the nearest tenth, the airplane’s horizontal distance from the raft is 6.2 km. Chapter 8 216 3 tan 268 5 d Copyright © by Pearson Education, Inc. or its affiliates. All Rights Reserved. 18. Label the diagram below. Lesson Check • Do you UNDERSTAND? Vocabulary How is an angle of elevation formed? Underline the correct word(s) to complete each sentence. 22. The angle of elevation is formed above / below a horizontal line. 23. The angle of depression is formed above / below a horizontal line. 24. The measure of an angle of elevation is equal to / greater than / less than the measure of the angle of depression. Lesson Check • Do you UNDERSTAND? Error Analysis A homework question says that the angle of depression from the bottom of a house window to a ball on the ground is 20°. At the right is your friend’s sketch of the situation. Describe your friend’s error. 20˚ 25. Is the angle that your friend identified as the angle of depression formed by the horizontal and the line of sight? Yes / No 26. Is the correct angle of depression adjacent to or opposite the angle identified by your friend? adjacent to / opposite Copyright © by Pearson Education, Inc. or its affiliates. All Rights Reserved. 27. Describe your friend’s error. Answers may vary. Sample: My friend identified the wrong angle. The _______________________________________________________________________ correct angle of depression is below the horizontal line. _______________________________________________________________________ _______________________________________________________________________ Math Success Check off the vocabulary words that you understand. angle of elevation angle of depression trigonometric ratios Rate how well you can use angles of elevation and depression. Need to review 0 2 4 6 8 Now I get it! 10 217 Lesson 8-4 Law of Sines 8-5 Vocabulary Review 1. Draw a line segment from each angle of the triangle to its opposite side. C b a A B c 2. Circle the correct word. A ratio is the comparison of two quantities by addition subtraction multiplication division sine (noun) syn Related Words: triangle, side length, angle measure, opposite, cosine Definition: In a right triangle, sine is the ratio of the side opposite a given acute angle to the hypotenuse. Example: If you know the measure of an acute angle of a right triangle and the length of the opposite side, you can use the sine ratio to find the length of the hypotenuse. Use Your Vocabulary 3. A triangle has a given acute angle. Circle its sine ratio. adjacent hypotenuse hypotenuse opposite opposite hypotenuse opposite adjacent 4. A right triangle has one acute angle measuring 36.98. The length of the side adjacent to this angle is 4 units, and the length of the side opposite this angle is 3 units. The length of the hypotenuse is 5 units. Circle the sine ratio of the 36.98 angle. 43 Chapter 8 3 5 4 5 5 4 5 3 218 Copyright © by Pearson Education, Inc. or its affiliates. All Rights Reserved. Vocabulary Builder Law of Sines For any nABC, let the lengths of the sides opposite angles A, B, and C be a, b, and c, respectively. C Then the Law of Sines relates the sine of each angle to the length of its opposite side. sin B sin C sin A a 5 b 5 c b 5. If you know 2 angles and 1 side of a triangle, can you find all of the missing measures? Explain. a A B c Yes; since the sum of a triangle’s angles are 1808, you can find the third angle; then use the Law of Sines to find the other 2 sides. Problem 1 Using the Law of Sines (AAS) Got It? In DABC, m/A 5 48, m/B 5 93, and AC 5 15. What is AB to the nearest tenth? C 398 6. Find and label m/C. 1808 5 488 1 938 1 39 c 7. Label side lengths a, b, and c. Which side is the length of AB? __________ a 8. Circle the equation which can be used to solve this problem. Explain your reasoning. sin C sin A c 5 a sin C sin B c 5 b 93 B sin B sin A 5 a b 15 b 48 A c Copyright © by Pearson Education, Inc. or its affiliates. All Rights Reserved. Answers may vary. Sample: I want to pick the equation in which ______________________________________________________________ I have 3 out of the 4 values. 9. Replace the variables in the equation with values from DABC. sin 398 c 5 sin 938 15 10. Find the sine values of the given angles, cross multiply, then solve for c. c< .6293 ) • ( 15 ) 9.453 ( < ( .9986 ) 9.5 units. 11. The length of AB is about Problem 2 Using the Law of Sines (SSA) L Got It? In DKLM , LM 5 9, KM 5 14, and m/L 5 105. m To the nearest tenth, what is m/K ? 12. Label the triangle with information from the problem and the length of the sides as k, l, m. 219 K 105 14 5 l 95k M Lesson 8-5 13. Use the letter that represents the length of KM to write a pair of ratios using some of the letters k, l, m, K, L and M. sin L l 5 sin K k 14. Fill in the values in the equation from Exercise 13 and solve for sin K. sin14105 5 sin9 K sin K < (.9659)(9) 14 < .6210 .6210 sin K < 15. Use your calculator and take the inverse sine of both sides of the equation to find m/K . sin 21 .6210 , therefore m/K < (sin K) < sin 21 38.4° Problem 3 Using the Law of Sines to Solve a Problem 16. Underline the correct word to complete each sentence. 2nd Base In this problem, the solution is a side / angle . 60 ft 17. In order to use the Law of Sines what information will you need that is missing and why? I need to find the measure of the angle formed by second base, ______________________________________________________________ the right fielder, and first base because I need the measure of ______________________________________________________________ the angle opposite the given side to use the Law of Sines. ______________________________________________________________ 18. Circle the equation you could use to solve for the missing solution. sin 688 sin 728 60 5 a sin 728 sin 408 c 60 5 sin 728 sin 688 60 5 b 19. Fill in the blanks to complete the equation. Then solve the equation and find the solution. 728 sin 60 0.9511 60 5 5 408 sin c 0.6428 c c (0.9511) < 60(0.6428) 40.6 feet. Kimmy throws the ball about Chapter 8 72 68 To find the solution, I need to first find a missing side / angle . Right-fielder a 220 40.6 c < b c 40 1st Base Copyright © by Pearson Education, Inc. or its affiliates. All Rights Reserved. Got It? The right-fielder fields a softball between first base and second base as shown in the figure. If the right-fielder throws the ball to second base, how far does she throw the ball? Lesson Check • Do you UNDERSTAND? Reasoning If you know the three side lengths of a triangle, can you use the Law of Sines to find the missing angle measures? Explain. 20. What do AAS, ASA, and SSA stand for? Match each term with its definition. Then tell what the three terms have in common. AAS Side-Side-Angle ASA Angle-Angle-Side SSA Angle-Side-Angle They all include sides and angles. ______________________________________________________________ 21. If you know only the three side lengths of a triangle, can you use the Law of Sines to find the missing angle measures? Explain. Answers may vary. Sample: No; To use the Law of Sines, you ______________________________________________________________ need 2 angles and 1 side or 2 sides and 1 angle. ______________________________________________________________ Error Analysis In DPQR, PQ 5 4 cm, QR 5 3 cm, and m/R 5 75. Your friend uses the Law of Sines to write Explain the error. sin 758 sin x8 5 4 to find m/Q. 3 P 22. Label the diagram with the given information. Did your friend correctly match the angles and the sides? Copyright © by Pearson Education, Inc. or its affiliates. All Rights Reserved. 4 Answers may vary. Sample: The side length opposite /R is 4, ________________________________________________________________________ not 3. /Q is not opposide side length 4. ________________________________________________________________________ 75 R x 3 Q Math Success Check off the vocabulary words that you understand. Law of Sines ratio adjacent inverse sine Rate how well you can use the Law of Sines. Need to review 0 2 4 6 8 Now I get it! 10 221 Lesson 8-5 Law of Cosines 8-6 Vocabulary Review Look at DABC. C b a A B c b and c 1. Name the sides that are adjacent to angle A. ___________ b 2. Which side is opposite of angle B? ______ 3. Identify each angle measure as acute, right, or obtuse. 458 acute ________ 1008 obtuse ________ 908 right ________ Cosine (noun) KOH syn Related Word: triangle, side length, angle measure, opposite, sine Definition: In a right triangle, cosine is the ratio of the side adjacent to a given acute angle to the hypotenuse. Example: If you know the measure of an acute angle of a right triangle and the length of the adjacent side, you can use the cosine ratio to find the length of the hypotenuse. Use Your Vocabulary 4. A triangle has a given acute angle. Circle its cosine ratio. adjacent hypotenuse hypotenuse adjacent opposite hypotenuse adjacent opposite 5. A right triangle has one acute angle measuring 53.18, the length of the side adjacent to this angle is 9 units, and the length of the side opposite this angle is 12 units. The length of the hypotenuse is 15 units. Circle the cosine ratio of the 53.18 angle. 9 12 Chapter 8 12 15 9 15 15 9 15 12 222 Copyright © by Pearson Education, Inc. or its affiliates. All Rights Reserved. Vocabulary Builder Law of Cosines For any nABC with side lengths a, b, and c opposite angles A, B, and C, respectively, the Law of Cosines relates the measures of the triangles according to the following equations. a2 5 b2 1 c2 2 2bc cos A b2 5 a2 1 c2 2 2ac cos B c2 5 a2 1 b2 2 2ab cos C C b a A d 2 5 f 2 1 e 2 2 2de cos D f 2 5 d 2 1 e 2 2 2de cos F e 2 5 d 2 1 f 2 2 df cos E B c 6. Circle the equation that is true for DDEF . E d f F D e Problem 1 Using the Law of Cosines (SAS) l M Got It? In DLMN, m/L 5 1048, LM 5 48, and LN 5 29. Find MN to N the nearest tenth. 48 n 7. Label the sides of DLMN with the letters l, m, and n. 104 29 m L 8. Use the information in the problem to complete the problem-solving model below. Know Need Copyright © by Pearson Education, Inc. or its affiliates. All Rights Reserved. LM is opposite / N MN 5 letter I n LM 5 48 5 letter LN is opposite / M LN 5 29 5 letter Plan m An equation using letters l, m, n, and /L. Because you know m/ L and need MN, substitute the angle measure and the two side lengths into the equation and solve for l. 9. Find MN by solving for l. a. I 2 5 m 2 1 n 2 2 2( m )( n ) cos /L a. Write an equation using l, m, n, and /L. b. l 2 5 29 2 1 48 2 2 2( 29 )( 48 ) cos 104 b. Substitute the values from the triangle. < 841 1 2304 2 (2784 ? 20.242) c. l2 c. Use the Order of Operations and solve for l 2 . < 3145 1 673.728 l2 l2 < 3818.728 d. l < 61.8 5 MN d. Take the square root of both sides. 223 Lesson 8-6 Problem 2 Using the Law of Cosines (SSS) T u Got It? In DTUV above, find m/T to the nearest tenth of a degree. 10. Label the sides of the triangle with t, u, and r. U 11. Solve for m/T following the given STEPS. 6.7 4.4 v t 7.1 V t 2 5 u 2 1 v 2 2 2( u )( v ) cos T Write an equation using the Law of Cosines. 7.1 2 5 6.7 2 1 4.4 2 2 2( 6.7 )( 4.4 ) cos T Substitute the values from the triangle. 50.41 5 44.89 1 19.36 Simplify by squaring and multiplying. 2 58.96 cos T 50.41 50.41 2 64.25 5 2 58.96 cos T Get coefficient of cos T and cos T alone. 213.84 58.96 cos 21 0.234735 Add the first two numbers. 5 cos T Divide by the coefficient of cos T. < T Take the inverse cosine of both sides of the equation. m/T < 76.4 Problem 3 Using the Law of Cosines to Solve a Problem Got It? You and a friend hike 1.4 miles due west from a campsite. At the same time two other friends hike 1.9 miles at a heading of S 118W (118west of south) from the campsite. To the nearest tenth of a mile, how far apart are the two groups? 12. Label the model with information from the problem and letter the angles and sides. West campsite c b = 1.9 miles a = 1.4 miles 11 South 13. Find the measure of the angle that is the complement of the 118angle. 798 908 2 118 5 Chapter 8 224 Copyright © by Pearson Education, Inc. or its affiliates. All Rights Reserved. 5 64.25 2 58.96 cos T 14. Write and solve an equation for finding the distance between the two groups. c2 5 1.42 1 1.92 2 2(1.4)(1.9) cos(79) c2 5 1.96 1 3.61 2 5.32 (.191) c2 5 5.57 2 1.02 c2 5 4.55 c < 2.13 miles Lesson Check • Do you UNDERSTAND? Writing Explain how you choose between the Law of Sines and the Law of Cosines when finding the measure of a missing angle or side. 15. Write C if you would use the Law of Cosines to find a missing measure in a triangle or S if you would use the Law of Sines. C The lengths of two sides and the measure of the included angle are given. Find the length of the third side. C The lengths of three sides are given. Find the measure of one angle. S The measures of two angles and the length of the included side are given. Find the length of another side. Copyright © by Pearson Education, Inc. or its affiliates. All Rights Reserved. 16. Explain how to choose between the Law of Sines and the Law of Cosines in solving a triangle. Answers may vary. Sample: Use the Law of Cosines when you know SAS ________________________________________________________________________ or SSS. Use the Law of Sines when you know SSA or ASA. ________________________________________________________________________ Math Success Check off the vocabulary words that you understand. Law of Cosines Law of Sines trigonometry Rate how well you can use the Law of Cosines. Need to review 0 2 4 6 8 Now I get it! 10 225 Lesson 8-6

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