792 CHAPTER 8 Applications of Trigonometry EXAMPLE 5 Using Heron’s Formula to Find an Area (SSS) The distance “as the crow flies” from Los Angeles to New York is 2451 mi, from New York to Montreal is 331 mi, and from Montreal to Los Angeles is 2427 mi. What is the area of the triangular region having these three cities as vertices? (Ignore the curvature of Earth.) SOLUTION In Figure 18, we let a = 2451, b = 331, and c = 2427. Montreal Los Angeles New York NOT TO SCALE c = 2427 mi a = 2451 mi b = 331 mi Figure 18 First, find the semiperimeter s. s = 1 2 1a + b + c2 Semiperimeter s = 1 2 12451 + 331 + 24272 Substitute the given values. s = 2604.5 Add, and then multiply. Now use Heron’s formula to find the area . = 2s1s - a21s - b21s - c2 = 22604.512604.5 - 2451212604.5 - 331212604.5 - 24272 ≈401,700 mi2 Use a calculator. Don’t forget the factor s. Derivation of Heron’s Formula A trigonometric derivation of Heron’s formula illustrates some ingenious manipulation. Let triangle ABC have sides of lengths a, b, and c. Apply the law of cosines. a2 = b2 + c2 - 2bc cos A Law of cosines cos A = b2 + c2 - a2 2bc Solve for cos A. (1) The perimeter of the triangle is a + b + c, so half of the perimeter (the semiperimeter) is given by the formula in equation (2) below. s = 1 2 1a + b + c2 (2) 2s = a + b + c Multiply by 2. (3) b + c - a = 2s - 2a Subtract 2a from each side and rewrite. b + c - a = 21s - a2 Factor. (4) Subtract 2b and 2c in a similar way in equation (3) to obtain the following. a - b + c = 21s - b2 (5) a + b - c = 21s - c2 (6) S Now Try Exercise 73.
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