Mock AIME 3 Pre 2005 Problems/Problem 9

Problem

$ABC$ is an isosceles triangle with base $\overline{AB}$ . $D$ is a point on $\overline{AC}$ and $E$ is the point on the extension of $\overline{BD}$ past $D$ such that $\angle{BAE}$ is right. If $BD = 15, DE = 2,$ and $BC = 16$ , then $CD$ can be expressed as $\frac{m}{n}$ , where $m$ and $n$ are relatively prime positive integers. Determine $m + n$ .

Solution

Let AB=x. Call the foot of the perpendicular from D to AB N, and the foot of the perpendicular from C to AB M. By similarity, AN=2x/17. Also, AM=x/2. Since $\triangle$ AND and $\triangle$ CAM are similar, we have (2x/17)/AD=(x/2)/16. Hence, AD=64/17, and CD=16-AD=208/17, so the answer is 225.

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Mock AIME 3 Pre 2005 Problems/Problem 9

Problem

Solution

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