Competition problem — Geometry — Maths Olympiad Prep

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Geometry Difficulty 4.5 AIME Find the answer

Let $A B C D$ be a square of side length 5. A circle passing through $A$ is tangent to segment $C D$ at $T$ and meets $A B$ and $A D$ again at $X \neq A$ and $Y \neq A$ , respectively. Given that $X Y=6$ , compute $A T$ .

Your answer

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

Solution

Let $O$ be the center of the circle, and let $Z$ be the foot from $O$ to $A D$ . Since $X Y$ is a diameter, $O T=Z D=3$ , so $A Z=2$ . Then $O Z=\sqrt{5}$ and $A T=\sqrt{O Z^{2}+25}=\sqrt{30}$ .

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