HMMT February

Geometry Difficulty 5.0 AIME, harder Prove it United States

Problem:

Triangle $G R T$ has $G R=5$ , $R T=12$ , and $G T=13$ . The perpendicular bisector of $G T$ intersects the extension of $G R$ at $O$ . Find $T O$ .

Solution

Solution:

First, note that $T O = G O$ as $O$ lies on the perpendicular bisector of $G T$ . Then if $M$ is the midpoint of $G T$ , we have that $\triangle G R T \sim \triangle G M O$ , so we can compute

$T O = G O = G M \cdot \frac{G T}{G R} = \frac{13}{2} \cdot \frac{13}{5} = \frac{169}{10}.$

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