Competition problem — Geometry — Maths Olympiad Prep

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Geometry Difficulty 4.1 AIME Find the answer United States

Problem:

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. Fractions can be typed as 3/2, and spacing doesn't matter.

Solution

Solution:

The radius is $\frac{25}{8}$ .

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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