In the diagram, pentagon $P Q R S T$ has $P Q=13$, $Q R=18, S T=30$, and a perimeter of 82 . Also, $\angle Q R S=\angle R S T=\angle S T P=90^{\circ}$. The area of the pentagon $P Q R S T$ is
(A) 306
(B) 297
(D) 279
(E) 270
(C) 288

![](https://cdn.mathpix.com/cropped/2024_04_20_e85ef69091c57cb633f3g-021.jpg?height=334&width=526&top_left_y=1107&top_left_x=1190)

Part B: Each correct answer is worth 6.
We extend $R Q$ to the left until it meets $P T$ at point $U$, as shown.

![](https://cdn.mathpix.com/cropped/2024_04_20_688a3b42f69739f2e244g-040.jpg?height=337&width=526&top_left_y=875&top_left_x=862)

Because quadrilateral $U R S T$ has three right angles, then it must have four right angles and so is a rectangle.

Thus, $U T=R S$ and $U R=T S=30$.

Since $U R=30$, then $U Q=U R-Q R=30-18=12$.

Now $\triangle P Q U$ is right-angled at $U$.

By the Pythagorean Theorem, since $P U>0$, we have

$$
P U=\sqrt{P Q^{2}-U Q^{2}}=\sqrt{13^{2}-12^{2}}=\sqrt{169-144}=\sqrt{25}=5
$$

Since the perimeter of $P Q R S T$ is 82 , then $13+18+R S+30+(U T+5)=82$.

Since $R S=U T$, then $2 \times R S=82-13-18-30-5=16$ and so $R S=8$.

Finally, we can calculate the area of $P Q R S T$ by splitting it into $\triangle P Q U$ and rectangle $U R S T$.

The area of $\triangle P Q U$ is $\frac{1}{2} \times U Q \times P U=\frac{1}{2} \times 12 \times 5=30$.

The area of rectangle $U R S T$ is $R S \times T S=8 \times 30=240$.

Therefore, the area of pentagon $P Q R S T$ is $30+240=270$.

ANSWER: (E)