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Unformatted text preview: Cosines to ﬁnd the length of the missing side opposite the given angle. Calling
this length w (for width ), we get w2 = 9502 + 10002 − 2(950)(1000) cos (60◦ ) = 952500 from which
we get w = 952500 ≈ 976 feet.
In Section 11.2, we used the proof of the Law of Sines to develop Theorem 11.4 as an alternate
formula for the area enclosed by a triangle. In this section, we use the Law of Cosines to prove
Heron’s Formula – a formula which computes the area enclosed by a triangle using only the lengths
of its sides.
Theorem 11.6. Heron’s Formula: Suppose a, b and c denote the lengths of the three sides of
a triangle. Let s be the semiperimeter of the triangle, that is, let s = 2 (a + b + c). Then the area
A enclosed by the triangle is given by
A= s(s − a)(s − b)(s − c) We begin proving Theorem 11.6 using Theorem 11.4. Using the convention that the angle γ is
opposite the side c, we have A = 1 ab sin(γ ) from Theorem 11.4. In order to simplify computations,
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