In the diagram, square $P Q R S$ has side length 2. Points $W, X, Y$, and $Z$ are the midpoints of the sides of $P Q R S$. What is the ratio of the area of square $W X Y Z$ to the area of square $P Q R S$ ?
(A) $1: 2$
(B) $2: 1$
(C) $1: 3$
(D) $1: 4$
(E) $\sqrt{2}: 2$

![](https://cdn.mathpix.com/cropped/2024_04_20_ac36362783317e0251fdg-045.jpg?height=371&width=379&top_left_y=579&top_left_x=1277)

Part B: Each correct answer is worth 6.
Solution 1

Since square $P Q R S$ has side length 2, then $P Q=Q R=R S=S P=2$.

Since $W, X, Y, Z$ are the midpoints of the sides of $P Q R S$, then $P W=P Z=1$.

Since $\angle Z P W=90^{\circ}$, then $W Z=\sqrt{P W^{2}+P Z^{2}}=\sqrt{1^{2}+1^{2}}=\sqrt{2}$.

Therefore, square $W X Y Z$ has side length $\sqrt{2}$.

The area of square $W X Y Z$ is $(\sqrt{2})^{2}=2$ and the area of square $P Q R S$ is $2^{2}=4$.

The ratio of these areas is $2: 4$ or $1: 2$.

Solution 2

Join $W$ to $Y$ and $X$ to $Z$.

![](https://cdn.mathpix.com/cropped/2024_04_20_33a8a6559d675f59baa2g-087.jpg?height=377&width=374&top_left_y=1340&top_left_x=930)

Since $P Q R S$ is a square and $W, X, Y$, and $Z$ are the midpoints of its sides, then $W Y$ and $Z X$ divide the square into four identical squares.

Each of these four squares is divided into two triangles of equal area by its diagonal. (These diagonals are $W Z, W X, X Y, Y Z$.)

Square $W X Y Z$ is made up of 4 of these triangles of equal area.

Square $P Q R S$ is made up of 8 of these triangles of equal area.

Therefore, the ratio of these areas is $4: 8$ or $1: 2$.

ANswer: (A)