800 CHAPTER 8 Applications of Trigonometry Relating Concepts For individual or collaborative investigation (Exercises 77 – 80) We have introduced two new formulas for the area of a triangle in this chapter. We can now find the area of a triangle using one of three formulas. (a) = 1 2 bh (b) = 1 2 ab sin C Aor = 1 2 ac sin B or = 1 2 bc sin AB (c) = 2s1s - a21s - b21s - c2 (Heron’s formula) If the coordinates of the vertices of a triangle are given, then the following area formula is also valid. (d) = 1 22 1x1y2 - y1x2 + x2 y3 - y2x3 + x3y1 - y3 x122 Work Exercises 77 – 80 in order, showing that the formulas give the same area. 77. Draw a triangle with vertices A12, 52, B1-1, 32, and C14, 02, and use the distance formula to find the lengths of the sides a, b, and c. 78. Find the area of triangle ABC using formula (b). (First use the law of cosines to find the measure of an angle.) 79. Find the area of triangle ABC using formula (c) — that is, Heron’s formula. 80. Find the area of triangle ABC using new formula (d). The vertices are the ordered pairs 1x1, y12, 1x2, y22, and 1x3, y32. Find the indicated part of each triangle ABC. 1. Find A if B = 30.6°, b = 7.42 in., and c = 4.54 in. 2. Find a if A = 144°, c = 135 m, and b = 75.0 m. 3. Find C if a = 28.4 ft, b = 16.9 ft, and c = 21.2 ft. Solve each problem. 4. Find the area of the triangle shown here. 9 7 1508 5. Find the area of triangle ABC if a = 19.5 km, b = 21.0 km, and c = 22.5 km. 6. For triangle ABC with c = 345, a = 534, and C = 25.4°, there are two possible values for angle A. What are they? 7. Solve triangle ABC if c = 326, A = 111°, and B = 41.0°. 8. Height of a Balloon The angles of elevation of a hot air balloon from two observation points X and Y on level ground are 42° 10′ and 23° 30′, respectively. As shown in the figure, points X, Y, and Z are in the same vertical plane and points X and Y are 12.2 mi apart. Find the height of the balloon to the nearest tenth of a mile. Chapter 8 Quiz (Sections 8.1 – 8.2) 12.2 mi 238 309 428 109 Z Y X
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