Unformatted text preview: in the following example. Example 2.9 c d a b a b C D Figure 2.2.2 Use the Law of Cosines to prove that the sum of the squares of the diagonals of any parallelogram equals the sum of the squares of the sides. Solution: Let a and b be the lengths of the sides, and let the diagonals opposite the angles C and D have lengths c and d , respectively, as in igure 2.2.2. Then we need to show that c 2 + d 2 = a 2 + b 2 + a 2 + b 2 = 2( a 2 + b 2 ) . By the Law of Cosines, we know that c 2 = a 2 + b 2 2 ab cos C , and d 2 = a 2 + b 2 2 ab cos D . By properties of parallelograms, we know that D = 180 C , so d 2 = a 2 + b 2 2 ab cos (180 C ) = a 2 + b 2 + 2 ab cos C , since cos (180 C ) = cos C . Thus, c 2 + d 2 = a 2 + b 2 2 ab cos C + a 2 + b 2 + 2 ab cos C = 2( a 2 + b 2 ) . QED...
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 Fall '11
 Dr.Cheun
 Calculus, Angles, Sets, Law Of Cosines, Law of sines, Cos, triangle, a2 + b2

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