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Problem 316

Geometry Difficulty 1.9 Find the answer CEMC Pascal

In rectangle $PQRS$ , $PS=6$ and $SR=3$ . Point $U$ is on $QR$ with $QU=2$ . Point $T$ is on $PS$ with $\angle TUR=90^{\circ}$ . What is the length of $TR$ ?

Your answer

A number or a short expression. Fractions can be typed as 3/2, and spacing doesn't matter.

Official solution

Since $PQRS$ is a rectangle, then $QR=PS=6$ . Therefore, $UR=QR-QU=6-2=4$ . Since $PQRS$ is a rectangle and $TU$ is perpendicular to $QR$ , then $TU$ is parallel to and equal to $SR$ , so $TU=3$ . By the Pythagorean Theorem, since $TR>0$ , then $TR=\sqrt{TU^{2}+UR^{2}}=\sqrt{3^{2}+4^{2}}=\sqrt{25}=5$ . Thus, $TR=5$ .

Source: Omni-MATH, licensed Apache-2.0. Statement reproduced verbatim; metadata (topic, difficulty, ordering) added by this project.