In the diagram, $P Q R S$ is a square and $M$ is the midpoint of $P Q$. The area of triangle $M Q R$ is 100 . The area of the square $P Q R S$ is
(A) 200
(B) 500
(C) 300
(D) 400
(E) 800

![](https://cdn.mathpix.com/cropped/2024_04_20_ac36362783317e0251fdg-068.jpg?height=322&width=331&top_left_y=1449&top_left_x=1298)
Solution 1

Draw a line from $M$ to $T$ on $S R$ so that $M T$ is parallel to $Q R$.

Then $M T R Q$ is a rectangle. This means that the area of $\triangle M Q R$

is half of the area of rectangle $M T R Q$.

Thus, the area of $M T R Q$ is $2 \times 100=200$.

Since $M$ is the midpoint of $P Q$ and $P Q R S$ is a square, then $T$ is the midpoint of $S R$.

This means that the area of $M T R Q$ is half of the area of $P Q R S$.

![](https://cdn.mathpix.com/cropped/2024_04_20_33a8a6559d675f59baa2g-136.jpg?height=328&width=350&top_left_y=1039&top_left_x=1538)

Therefore, the area of $P Q R S$ is $2 \times 200=400$.

Solution 2

Suppose that the side length of square $P Q R S$ is $2 x$.

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

Since $P Q R S$ is a square, then $\triangle M Q R$ is right-angled at $Q$.

Therefore, the area of $\triangle M Q R$ is $\frac{1}{2}(M Q)(Q R)=\frac{1}{2}(x)(2 x)=x^{2}$.

Since the area of $\triangle M Q R$ is 100 , then $x^{2}=100$, and so $x=10$, since $x>0$.

Thus, the side length of square $P Q R S$ is $2 x=20$ and so the area of square $P Q R S$ is $20^{2}=400$.

ANSWER: (D)