799 8.2 The Law of Cosines Find the exact area of each triangle using the formula = 1 2 bh, and then verify that Heron’s formula gives the same result. 63. 6 14 16 3 3 64. 6 14 10 3 3 Find the area of each triangle ABC. See Example 5. 65. a = 12 m, b = 16 m, c = 25 m 66. a = 22 in., b = 45 in., c = 31 in. 67. a = 154 cm, b = 179 cm, c = 183 cm 68. a = 25.4 yd, b = 38.2 yd, c = 19.8 yd 69. a = 76.3 ft, b = 109 ft, c = 98.8 ft 70. a = 15.8 m, b = 21.7 m, c = 10.9 m Solve each problem. See Example 5. 71. Perfect Triangles A perfect triangle is a triangle whose sides have whole number lengths and whose area is numerically equal to its perimeter. Show that the triangle with sides of length 9, 10, and 17 is perfect. 72. Heron Triangles A Heron triangle is a triangle having integer sides and area. Show that each of the following is a Heron triangle. (a) a = 11, b = 13, c = 20 (b) a = 13, b = 14, c = 15 (c) a = 7, b = 15, c = 20 (d) a = 9, b = 10, c = 17 73. Area of the Bermuda Triangle Find the area of the Bermuda Triangle, to the nearest thousand square miles, if the sides of this triangle measure approximately 960 mi, 1030 mi, and 1030 mi. Bermuda 1030 mi 1030 mi 960 mi Miami San Juan Puerto Rico Caribbean Sea Bermuda Triangle 74. Required Amount of Paint A painter needs to cover a triangular region 75 m by 68 m by 85 m. A can of paint covers 75 m2 of area. How many cans (to the next higher number of cans) will be needed? 75. Consider triangle ABC shown here. (a) Use the law of sines to find candidates for the value of angle C. Round angle measures to the nearest tenth of a degree. (b) Rework part (a) using the law of cosines. (c) Why is the law of cosines a better method in this case? 76. Show that the measure of angle A in triangle ABC shown here is twice the measure of angle B. (Hint: Use the law of cosines to find cos A and cos B, and then show that cos A = 2 cos2 B - 1.) A C B a c b 7 13 15 608 A B C 5 6 4
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