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Geometry Difficulty 4.8 AIME Find the answer United States

Problem:

Let triangle $ABC$ be such that $AB = AC = 22$ and $BC = 11$ . Point $D$ is chosen in the interior of the triangle such that $AD = 19$ and $\angle ABD + \angle ACD = 90^\circ$ . The value of $BD^2 + CD^2$ can be expressed as $\frac{a}{b}$ , where $a$ and $b$ are relatively prime positive integers. Compute $100a + b$ .

Your answer

A number or a short expression. Fractions can be typed as 3/2, and spacing doesn't matter.

Solution

Solution:

Rotate triangle $ABD$ about $A$ so that $B$ coincides with $C$ . Let $D$ map to $D'$ under this. Note that $CDD'$ is a right triangle with right angle at $C$ . Also, note that $ADD'$ is similar to $ABC$ . Thus, we have $DD' = \frac{AD}{2} = \frac{19}{2}$ . Finally, note that
$BD^2 + CD^2 = CD'^2 + CD^2 = DD'^2 = \frac{361}{4}$

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Source: MathNet, licensed CC-BY-4.0. Statement reproduced verbatim; metadata (topic, difficulty) added by this project.