In the diagram, $P Q R S$ is a rectangle with $S R=15$. Point $T$ is above $P S$ and point $U$ is on $P S$ so that $T U$ is perpendicular to $P S$. If $P T=10$ and $U S=4$ and the area of $P Q R S$ is 180 , what is the area of $\triangle P T S ?$
(A) 60
(B) 36
(C) 48
(D) 24
(E) 12

![](https://cdn.mathpix.com/cropped/2024_04_20_ac36362783317e0251fdg-032.jpg?height=480&width=358&top_left_y=2118&top_left_x=1293)
Since the area of rectangle $P Q R S$ is 180 and $S R=15$, then $P S=\frac{180}{15}=12$.

Since $P S=12$ and $U S=4$, then $P U=P S-U S=12-4=8$.

Since $\triangle P U T$ is right-angled at $U$, then by the Pythagorean Theorem,

$$
T U=\sqrt{P T^{2}-P U^{2}}=\sqrt{10^{2}-8^{2}}=\sqrt{36}=6
$$

since $T U>0$.

In $\triangle P T S$, we can consider base $P S$ and height $T U$.

Therefore, its area is $\frac{1}{2}(P S)(T U)=\frac{1}{2}(12)(6)=36$.

ANSWER: (B)