HMMT, 2014

Geometry Difficulty 4.9 AIME Prove it United States

Problem:

Point $P$ and line $\ell$ are such that the distance from $P$ to $\ell$ is $12$ . Given that $T$ is a point on $\ell$ such that $PT = 13$ , find the radius of the circle passing through $P$ and tangent to $\ell$ at $T$ .

Solution

Solution:

Answer: $\dfrac{169}{24}$

Let $O$ be the center of the given circle, $Q$ be the foot of the altitude from $P$ to $\ell$ , and $M$ be the midpoint of $PT$ . Then since $OM \perp PT$ and $\angle OTP = \angle TPQ$ , $\triangle OMP \sim \triangle TQP$ . Thus the $OP = TP \cdot \dfrac{PM}{PQ} = 13 \cdot \dfrac{13/2}{12} = \dfrac{169}{24}$

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