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Warm Up Find each measure to the nearest tenth. 1 . m  y 2. x 3. y

Warm Up Find each measure to the nearest tenth. 1 . m  y 2. x 3. y 4. What is the area of ∆ XYZ ? Round to the nearest square unit. ≈ 8.8. 104°. ≈ 18.3. 60 square units. Objectives. Use the Law of Cosines to find the side lengths and angle measures of a triangle.

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Warm Up Find each measure to the nearest tenth. 1 . m  y 2. x 3. y

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  1. Warm Up Find each measure to the nearest tenth. 1.my2. x 3. y 4. What is the area of ∆XYZ ? Round to the nearest square unit. ≈ 8.8 104° ≈ 18.3 60 square units

  2. Objectives Use the Law of Cosines to find the side lengths and angle measures of a triangle. Use Heron’s Formula to find the area of a triangle.

  3. Use the Law of Cosines for which side-angle-side (SAS) or side-side-side (SSS) information is given .

  4. Example 1A: Using the Law of Cosines Use the given measurements to solve ∆ABC. Round to the nearest tenth. a = 8, b = 5, mC = 32.2° Step 1 Find the length of the third side. c2 = a2 + b2– 2ab cos C Law of Cosines c2 = 82 + 52– 2(8)(5)cos 32.2° Substitute. c2 ≈ 21.3 Use a calculator to simplify. c ≈ 4.6 Solve for the positive value of c.

  5. Solve for m B. Example 1A Continued Step 2 Find the measure of the smaller angle, B. Law of Sines Substitute. Solve for sin B.

  6. Solve for m A. Example 1A Continued Step 3 Find the third angle measure. mA + 35.4° + 32.2°  180° Triangle Sum Theorem mA 112.4°

  7. Step 1 Find the measure of the largest angle, B. m B = Cos-1 (0.2857) ≈ 73.4° Solve for m B. Example 1B: Using the Law of Cosines Use the given measurements to solve ∆ABC. Round to the nearest tenth. a = 8, b = 9, c = 7 b2 = a2 + c2– 2ac cos B Law of cosines 92 = 82 + 72– 2(8)(7)cos B Substitute. cos B = 0.2857 Solve for cos B.

  8. m C = Cos-1 (0.6667) ≈ 48.2° Solve for m C. Example 1B Continued Use the given measurements to solve ∆ABC. Round to the nearest tenth. Step 2 Find another angle measure c2 = a2 + b2– 2ab cos C Law of cosines 72 = 82 + 92– 2(8)(9)cos C Substitute. cos C = 0.6667 Solve for cos C.

  9. m A + 73.4° + 48.2°  180° m A 58.4° Solve for m A. Example 1B Continued Use the given measurements to solve ∆ABC. Round to the nearest tenth. Step 3 Find the third angle measure. Triangle Sum Theorem

  10. b = 23, c = 18, m A = 173° Check It Out! Example 1a Use the given measurements to solve ∆ABC. Round to the nearest tenth. Step 1 Find the length of the third side. a2 = b2 + c2– 2bc cos A Law of Cosines a2 = 232 + 182– 2(23)(18)cos 173° Substitute. a2 ≈ 1672.8 Use a calculator to simplify. a ≈ 40.9 Solve for the positive value of c.

  11. Solve for m C. m C = Sin-1 Check It Out! Example 1a Continued Step 2 Find the measure of the smaller angle, C. Law of Sines Substitute. Solve for sin C.

  12. m B + 3.1° + 173°  180° m B 3.9° Solve for m B. Check It Out! Example 1a Continued Step 3 Find the third angle measure. Triangle Sum Theorem

  13. Step 1 Find the measure of the largest angle, C. m C = Cos-1 (0.1560) ≈ 81.0° Solve for m C. Check It Out! Example 1b Use the given measurements to solve ∆ABC. Round to the nearest tenth. a = 35, b = 42, c = 50.3 c2 = a2 + b2– 2ab cos C Law of cosines 50.32 = 352 + 422– 2(35)(50.3) cos C Substitute. cos C = 0.1560 Solve for cos C.

  14. m A = Cos-1 (0.7264) ≈ 43.4° Solve for m A. Check It Out! Example 1b Continued Use the given measurements to solve ∆ABC. Round to the nearest tenth. a = 35, b = 42, c = 50.3 Step 2 Find another angle measure a2 = c2 + b2– 2cb cos A Law of cosines 352 = 50.32 + 422– 2(50.3)(42) cos A Substitute. cos A = 0.7264 Solve for cos A.

  15. m B + 81° + 43.4°  180° m B 55.6° Solve for m B. Check It Out! Example 1b Continued Step 3 Find the third angle measure.

  16. Remember! The largest angle of a triangle is the angle opposite the longest side.

  17. The Law of Cosines can be used to derive a formula for the area of a triangle based on its side lengths. This formula is called Heron’s Formula.

  18. Example 3: Landscaping Application A garden has a triangular flower bed with sides measuring 2 yd, 6 yd, and 7 yd. What is the area of the flower bed to the nearest tenth of a square yard? Step 1 Find the value of s. Use the formula for half of the perimeter. Substitute 2 for a, 6 for b, and 7 for c.

  19. A = A = Example 3 Continued Step 2 Find the area of the triangle. Heron’s formula Substitute 7.5 for s. A = 5.6 Use a calculator to simplify. The area of the flower bed is 5.6 yd2.

  20. m C ≈ 112.0° Solve for m c. Find the area of the triangle by using the formula area = ab sin c. area Example 3 Continued Check Find the measure of the largest angle, C. c2 = a2+ b2– 2ab cos C Law of Cosines 72 = 22+ 62– 2(2)(6) cos C Substitute. Solve for cos C. cos C ≈ –0.375 

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