In the diagram, rectangle $P Q R S$ has $P S=6$ and $S R=3$. Point $U$ is on $Q R$ with $Q U=2$. Point $T$ is on $P S$ with $\angle T U R=90^{\circ}$. What is the length of $T R$ ?
(A) 3
(B) 4
(C) 5
(D) 6
(E) 7

![](https://cdn.mathpix.com/cropped/2024_04_20_6ed09463f225f8ba1f07g-063.jpg?height=440&width=499&top_left_y=257&top_left_x=1298)
Since $P Q R S$ is a rectangle, then $Q R=P S=6$.

Therefore, $U R=Q R-Q U=6-2=4$.

Since $P Q R S$ is a rectangle and $T U$ is perpendicular to $Q R$, then $T U$ is parallel to and equal to $S R$, so $T U=3$.

By the Pythagorean Theorem, since $T R>0$, then

$$
T R=\sqrt{T U^{2}+U R^{2}}=\sqrt{3^{2}+4^{2}}=\sqrt{25}=5
$$

Thus, $T R=5$. (We could also notice the 3-4-5 right-angled triangle.)

ANSWER: $(\mathrm{C})$