Pre-Calc Exam Notes 57

Pre-Calc Exam Notes 57 - The Area of a Triangle • Section...

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Unformatted text preview: The Area of a Triangle • Section 2.4 57 Example 2.15 b = 4 a = 5 c = 7 A B C Find the area of the triangle △ ABC given a = 5, b = 4, and c = 7. Solution: Using Heron’s formula with s = 1 2 ( a + b + c ) = 1 2 (5 + 4 + 7) = 8, the area K is given by: K = radicalbig s ( s − a )( s − b )( s − c ) = radicalbig 8(8 − 5)(8 − 4)(8 − 7) = radicallow 96 ⇒ K = 4 radicallow 6 ≈ 9.8 . Heron’s formula is useful for theoretical purposes (e.g. in deriving other formulas). How- ever, it is not well-suited for calculator use, exhibiting what is called numerical instability for “extreme” triangles, as in the following example. Example 2.16 Find the area of the triangle △ ABC given a = 1000000, b = 999999.9999979, and c = 0.0000029. Solution: To use Heron’s formula, we need to calculate s = 1 2 ( a + b + c ). Notice that the actual value of a + b + c is 2000000.0000008, which has 14 digits. Most calculators can store 12-14 digits internally (even if they display less), and hence may round off that value of...
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