To apply the Law of Cosines. Key Concept Law of Cosines

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1 Law of osines ontent Standards G.SRT.11 Understand and apply the... Law of osines... lso G.SRT.10 Ojective To apply the Law of osines c a MTHEMTIL PRTIES In the Solve It, you used right triangle trigonometry to write an epression to descrie a side length. You can also find relationships etween the angle measures and the side lengths of nonright triangles. Essential Understanding If you know the measures of two side lengths and the measure of the included angle (SS), or all three side lengths (SSS), then you can find all the other measures of the triangle. Key oncept Law of osines Lesson L VocaularyV Law of osines For any, the Law of osines relates the cosine of each angle to the side lengths of the triangle. a 2 2 c 2 2c cos 2 a 2 c 2 2ac cos c 2 a 2 2 2a cos a c Here s Why It Works Note that 2 2 h 2 and cos. Use the Pythagorean Theorem with D and simplify. a 2 (c ) 2 h 2 Pythagorean Theorem a 2 c 2 2c 2 h 2 Simplify. a 2 c 2 2c cos 2 Sustitute 2 for 2 h 2 and cos for. D h c a c a 2 2 c 2 2c cos ommutative Property -6 Law of osines 1

2 Prolem 1 Using the Law of osines (SS) Find to the nearest tenth. is opposite, so c 10. is opposite, so a 22. m 44 the length of ecause you know m and need, sustitute the angle measure and the two side lengths into 2 a 2 c 2 2ac cos and solve for a 2 c 2 2ac cos Law of osines (22)(10) cos The value of is aout Sustitute. Use a calculator. M N Got It? 1. Find MN to the nearest tenth. L Prolem 2 Using the Law of osines (SSS) How can you use what you know to find m v? You know the three side lengths (SSS) so you can use the Law of osines to find m V. In TUV, TU 4.4, UV 7.1, and TV 6.7. Find m V to the nearest tenth of a degree. Step 1 Step 2 Draw and lael a diagram. Use the Law of osines to set up an equation. TU 2 UV 2 TV 2 2(UV)(TV) cos V (7.1)(6.7) cos V Step 3 Solve for m V. U 4.4 T V cos V Simplify cos V Solve for cos V V cos Solve for m V m V Use a caclulator. The measure of V is aout Got It? 2. In TUV aove, find m T to the nearest tenth degree. 2 ommon ore

3 You can use the Law of osines to solve real world prolems involving triangles. Prolem 3 Using the Law of osines to Solve a Prolem What do you need to find efore you can use the law of osines? You need to find the measure of the angle opposite d in the triangle efore you can apply the Law of osines. n air traffic controller is tracking a plane 2.1 kilometers due south of the radar tower. second plane is located 3.5 kilometers from the tower at a heading of N 75 E (75 east of north). To the nearest tenth of a kilometer, how far apart are the two planes? The north-south line in the figure represents a straight angle. Let the angle opposite d e D. Use supplementary angles to find m D. Use supplementary angles to find the measure of the angle opposite d. m D Supplementary angles Tower 2.1 km North 75 Plane 3.5 km d Plane Use the Law of osines to solve for d. d 2 a 2 2 2a cos D Law of osines d (3.5)(2.1) cos 105 Sustitute. d Use a calculator. The distance etween the two planes is aout 4.5 kilometers. Got It? 3. You and a friend hike 1.4 miles due west from a campsite. t the same time, two other friends hike 1.9 miles at a heading of S 11 W (11 west of south) from the campsite. To the nearest tenth of a mile, how far apart are the two groups? Lesson heck Do you know HOW? 1. In, 7, 10, and m 80. To the nearest tenth, what is? 2. In QRS, QR 31.9, RS 25.2, and QS To the nearest tenth, what is m R? 3. In LMN, LN 7, MN 10, and m N 48. To the nearest tenth, what is the area of LMN? 4. What are m X, m Y, and m Z? 4 m Y X 7 m 6 m Z Do you UNDERSTND? 5. Error nalysis In, 15 ft, 12 ft, and m 32. student solved for c for a 12 ft, 15 ft, and m 32. What was the error? c = (12)(15)cos32 c = cos32 c = 63.7 MTHEMTIL PRTIES 6. Reasoning Eplain how you would find the measure of the largest angle of a triangle if given the measures of the three side lengths. -6 Law of osines 3

4 Practice and Prolem-Solving Eercises MTHEMTIL PRTIES Practice Use the information given to solve. See Prolems 1 and In QRS, m R 38, QR 11, and RS 16. To the nearest tenth, what is the length of QS? 8. In WXY, WX 20.4, XY 16.4, and WY To the nearest tenth, what is m W? 9. In JKL, JK 2.6, KL 6.4, and m K To the nearest tenth, what is the length of JL? 10. In DEF, DE 13, EF 24, and FD 27. To the nearest tenth, what is m E? Use the Law of osines to find the values of and y. Round to the nearest tenth y 40 5 y y y Use the Law of osines to solve each prolem. See Prolem aseall fter fielding a ground all, a pitcher is located 110 feet from first ase and 57 feet from home plate as shown in the figure at the right. To the nearest tenth, what is the measure of the angle with its verte at the pitcher? 16. Zipline One side of a ravine is 14 ft long. The other side is 12 ft long. 20 ft zipline runs from the top of one side of the ravine to the other. To the nearest tenth, at what angle do the sides of the ravine meet? pitcher 57 ft 110 ft Home Plate 90 ft 1st ase 20 ft pply 12 ft 14 ft 17. Think out a Plan walking path around the outside of a garden is shaped like a triangle. Two sides of the path that measure 32 ft and 39 ft form a 76 angle. If you walk around the entire path one time, how far have you walked? Write your answer to the nearest foot. What information do you need to find efore you can solve this prolem? How can you find the information you need? an drawing a diagram help you solve this prolem? 4 ommon ore

5 18. irplane commuter plane flies from ity to ity, a distance of 90 mi due north. Due to ad weather, the plane is redirected at take-off to a heading N 60 W (60 west of north). fter flying 57 mi, the plane is directed to turn northeast and fly directly toward ity. To the nearest tenth, how many miles did the plane fly on the last leg of the trip? For each triangle shown elow, determine whether you would use the Law of Sines or Law of osines to find the value of. Then find the value of to the nearest tenth mi ity 90 mi 60 ity ft water slide has a 9.5-ft ladder which meets the slide at a 95 angle. To the nearest tenth, what is the distance etween the end of the slide and the ottom of the ladder? 9.5 ft ft hallenge 24. Flags The dimensions of a triangular flag are 18 ft y 25 ft y 27 ft. To the nearest tenth, what is the measure of the angle formed y the two shorter sides? 25. Parallelogram QRST has a perimeter of 62 mm. To the nearest tenth, what is the length of TR? Q R STEM T S 26. Surveying surveyor measures the distance to the ase of a monument to e 12.4 meters at an angle of elevation of 11. t an angle of elevation of 26, the distance to the top of the monument is 13.3 meters. What is the height of the monument to the nearest tenth? m m 11-6 Law of osines 5

6 27. n isosceles triangle XYZ has a ase of 12 in. and a height of 8 in. To the nearest tenth, what are the measures of the angles? 28. Open-Ended Descrie a situation in which you are given three measures of a triangle ut are unale to solve the triangle for the other three measures. Standardized Test Prep ST/T 29. The circumference of a circle is 24 mm. To the nearest tenth of a millimeter, what is the radius of the circle? Use 3.14 for. 30. The lengths of the legs of a right triangle are 6 and 11. To the nearest hundredth, what is the length of the hypotenuse? 31. The medians E, F, and D of intersect at G. If GD 4, what is D? 237 yd yd 32. surveyor starts at one end of the lake and walks 192 yd. Then he turns 110º and walks 237 yd until he arrives at the other end of the lake. To the nearest tenth of a yard, what is the distance across the lake? Mied Review 33. The first leg of a ike race is 6 km due east. For the second leg of the race, the riders turn northwest and ride 9 km. The final leg of the race runs at a 40 angle to the second leg and rings the racers ack to the starting point. To the nearest tenth, at what angle measure does the third leg of the race meet the first leg? 34. Is the quadrilateral with vertices ( 1, 5), (6, 5), (9, 3), and D(2, 3) a parallelogram? Eplain. See Lesson 8-4. See Lesson 6-7. Get Ready! To prepare for Lesson 9-1, do Eercises EFG. omplete the congruence statements EG F See Lesson FG E 40. G E 6 ommon ore