In the diagram, square $P Q R S$ has side length 40 . Points $J, K, L$, and $M$ are on the sides of $P Q R S$, as shown, so that $J Q=K R=L S=M P=10$. Line segments $J Z$, $K W, L X$, and $M Y$ are drawn parallel to the diagonals of the square so that $W$ is on $J Z, X$ is on $K W, Y$ is on $L X$, and $Z$ is on $M Y$. What is the area of quadrilateral $W X Y Z$ ?
(A) 280
(B) 200
(C) 320
(D) 240
(E) 160

![](https://cdn.mathpix.com/cropped/2024_04_20_6ed09463f225f8ba1f07g-028.jpg?height=428&width=420&top_left_y=1930&top_left_x=1297)

Part C: Each correct answer is worth 8.
Since the sides of quadrilateral $W X Y Z$ are parallel to the diagonals of square $P Q R S$ and the diagonals of a square are perpendicular, then the sides of $W X Y Z$ are themselves perpendicular. This means that quadrilateral $W X Y Z$ has four right angles and is thus a rectangle.

Since the diagram does not change when rotated by $90^{\circ}, 180^{\circ}$ or $270^{\circ}$, then it must be the case that $W X=X Y=Y Z=Y W$ which means that $W X Y Z$ is a square.

We calculate the area of $W X Y Z$ by first calculating the length of $W Z$.

Extend $K W$ to meet $P Q$ at $T$. Join $M$ to $T$.

![](https://cdn.mathpix.com/cropped/2024_04_20_faa9db06d997f2a5b9b1g-049.jpg?height=417&width=417&top_left_y=564&top_left_x=908)

Since $T K$ is parallel to diagonal $P R$, then $\angle Q T K=\angle Q K T=45^{\circ}$, which means that $\triangle T Q K$ is isosceles with $Q T=Q K$.

Since $Q R=40$ and $K R=10$, then $Q K=Q R-K R=30$ and so $Q T=30$.

Since $P Q=40$, then $P T=P Q-Q T=10$.

Since $P M=P T=10$, then $\triangle M P T$ is right-angled and isosceles as well, which means that $M T$ is actually parallel to diagonal $S Q$. (We did not construct $M T$ with this property, but it turns out to be true.)

Since the sides of $M T W Z$ are parallel to the diagonals of the square, then $M T W Z$ is also a rectangle, which means that $W Z=M T$.

Since $P M=P T$, then

$$
M T=\sqrt{P M^{2}+P T^{2}}=\sqrt{10^{2}+10^{2}}=\sqrt{200}
$$

by the Pythagorean Theorem.

Thus, $W Z=M T=\sqrt{200}$ and so the area of square $W X Y Z$ equals $W Z^{2}$ or 200 .

ANSWER: (B)