$P Q R S$ is a square. The midpoint of $P Q$ is $M$ and the midpoint of $R S$ is $N$. If the perimeter of rectangle $P M N S$ is 36 , the area of square $P Q R S$ is
(A) 81
(B) 72
(C) 324
(D) 144
(E) 36

Part B: Each correct answer is worth 6.
Suppose that the side length of square $P Q R S$ is $x$.

Then $P Q=Q R=R S=S P=x$.

Since $M$ is the midpoint of $P Q$, then $P M=\frac{1}{2} x$.

In terms of $x$, the perimeter of rectangle $P M N S$ is

$$
2(P M+P S)=2\left(\frac{1}{2} x+x\right)=3 x
$$

![](https://cdn.mathpix.com/cropped/2024_04_20_33a8a6559d675f59baa2g-123.jpg?height=347&width=355&top_left_y=951&top_left_x=1601)

(Note that $S N=P M=\frac{1}{2} x$ since $N$ is the midpoint of $R S$. Also, $M N=P S=x$, since $M N$ is parallel to $P S$ and joins two parallel line segments.)

Since we are told that the perimeter of $P M N S$ is 36 , then $3 x=36$ or $x=12$.

Therefore, the area of square $P Q R S$ is $x^{2}=144$.

ANSWER: (D)