Competition problem — Geometry — Maths Olympiad Prep

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Geometry Difficulty 5.0 AIME Find the answer

Given right triangle $ABC$ , with $AB=4, BC=3$ , and $CA=5$ . Circle $\omega$ passes through $A$ and is tangent to $BC$ at $C$ . What is the radius of $\omega$ ?

Your answer

A number or a short expression. Spacing and $ signs are ignored.

Solution

Let $O$ be the center of $\omega$ , and let $M$ be the midpoint of $AC$ . Since $OA=OC$ , $OM \perp AC$ . Also, $\angle OCM=\angle BAC$ , and so triangles $ABC$ and $CMO$ are similar. Then, $CO/CM=AC/AB$ , from which we obtain that the radius of $\omega$ is $CO=\frac{25}{8}$ .

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