2009 AMC 12B Problems/Problem 13

Problem

Triangle $ABC$ has $AB = 13$ and $AC = 15$ , and the altitude to $\overline{BC}$ has length $12$ . What is the sum of the two possible values of $BC$ ?

$\mathrm{(A)}\ 15\qquad \mathrm{(B)}\ 16\qquad \mathrm{(C)}\ 17\qquad \mathrm{(D)}\ 18\qquad \mathrm{(E)}\ 19$

Solution

Let $D$ be the foot of the altitude to $\overline BC$ . Then $BD = \sqrt {13^2 - 12^2} = 5$ and $DC = \sqrt {15^2 - 12^2} = 9$ . Thus $BC = BD + BC = 5 + 9 = 14$ or $BC = DC - BD = 9 -5 = 4$ . The sum of the two possible values is $14 + 4 = \boxed{18}$ . The answer is $\mathrm{(D)}$ .

2009 AMC 12B (Problems • Answer Key • Resources)
Preceded by Problem 12	Followed by Problem 14
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2009 AMC 12B Problems/Problem 13

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