Each of $\triangle P Q R$ and $\triangle S T U$ has an area of 1. In $\triangle P Q R$, $U, W$ and $V$ are the midpoints of the sides, as shown. In $\triangle S T U, R, V$ and $W$ are the midpoints of the sides. What is the area of parallelogram $U V R W$ ?
(A) 1
(B) $\frac{1}{2}$
(C) $\frac{1}{3}$
(D) $\frac{1}{4}$
(E) $\frac{2}{3}$

![](https://cdn.mathpix.com/cropped/2024_04_20_46ead6524a8d61e21c51g-062.jpg?height=371&width=334&top_left_y=454&top_left_x=1340)

## Part C: Each correct answer is worth 8.
Since $V$ is the midpoint of $P R$, then $P V=V R$.

Since $U V R W$ is a parallelogram, then $V R=U W$.

Since $W$ is the midpoint of $U S$, then $U W=W S$.

Thus, $P V=V R=U W=W S$.

Similarly, $Q W=W R=U V=V T$.

Also, $R$ is the midpoint of $T S$ and therefore, $T R=R S$.

Thus, $\triangle V T R$ is congruent to $\triangle W R S$, and so the two triangles have equal area.

Diagonal $V W$ in parallelogram $U V R W$ divides the area of the parallelogram in half.

Therefore, $\triangle U V W$ and $\triangle R W V$ have equal areas.

In quadrilateral $V R S W, V R=W S$ and $V R$ is parallel to $W S$.

Thus, $V R S W$ is a parallelogram and the area of $\triangle R W V$ is equal to the area of $\triangle W R S$.

Therefore, $\triangle V T R, \triangle W R S, \triangle R W V$, and $\triangle U V W$ have equal areas, and so these four triangles divide $\triangle S T U$ into quarters.

Parallelogram $U V R W$ is made from two of these four quarters of $\triangle S T U$, or one half of $\triangle S T U$. The area of parallelogram $U V R W$ is thus $\frac{1}{2}$ of 1 , or $\frac{1}{2}$.

ANSWER: (B)