Two rectangles $P Q U V$ and $W S T V$ overlap as shown. What is the area of $P Q R S T V$ ?
(A) 35
(B) 24
(C) 25
(D) 17
(E) 23

![](https://cdn.mathpix.com/cropped/2024_04_20_6ed09463f225f8ba1f07g-075.jpg?height=498&width=412&top_left_y=171&top_left_x=1301)
Solution 1

Since $P Q U V$ and $W S T V$ are rectangles that share a common right angle at $V$, then $P Q, W S$ and $V T$ are parallel, as are $P V, Q U$, and $S T$. This tells us that all of the angles in the diagram are right angles.

Since $P Q U V$ is a rectangle, then $V U=P Q=2$.

Since $V T=5$ and $V U=2$, then $U T=V T-V U=5-2=3$.

Note that $R S T U$ is a rectangle, since it has four right angles.

Therefore, the area of $P Q R S T V$ equals the sum of the areas of rectangles $P Q U V$ and $R S T U$, or $2 \times 7+3 \times 3=23$.

(We could also consider the area of $P Q R S T V$ to be the sum of the areas of rectangle $P Q R W$ and rectangle $W S T V$.)

## Solution 2

Since $P Q U V$ and $W S T V$ are rectangles that share a common right angle at $V$, then $P Q, W S$ and $V T$ are parallel, as are $P V, Q U$, and $S T$. This tells us that all of the angles in the diagram are right angles.

We can consider $P Q R S T V$ to be a large rectangle $P X T V$ with a smaller rectangle $Q X S R$ removed.

![](https://cdn.mathpix.com/cropped/2024_04_20_faa9db06d997f2a5b9b1g-127.jpg?height=506&width=415&top_left_y=636&top_left_x=909)

The area of rectangle $P X T V$ is $7 \times 5=35$.

Since $P Q U V$ is a rectangle, then $Q U=P V=7$.

Since $P V$ is parallel to $Q U$ and $S T$, then $R U=S T=3$.

Thus, $Q R=Q U-R U=7-3=4$.

Since $W S T V$ is a rectangle, then $W S=V T=5$.

Since $V T$ is parallel to $W S$ and $P Q$, then $W R=P Q=2$.

Thus, $R S=W S-W R=5-2=3$.

Therefore, rectangle $Q X S R$ is 4 by 3, and so has area 12 .

Therefore, the area of $P Q R S T V$ is $35-12=23$.

## Solution 3

Since $P Q U V$ and $W S T V$ are rectangles that share a common right angle at $V$, then $P Q, W S$ and $V T$ are parallel, as are $P V, Q U$, and $S T$. This tells us that all of the angles in the diagram are right angles.

If we add up the areas of rectangle $P Q U V$ and $W S T V$, we get exactly the region $P Q R S T V$, but have added the area of $W R U V$ twice. Thus, the area of $P Q R S T V$ equals the area of $P Q U V$ plus the area of $W S T V$ minus the area of $W R U V$.

We note that rectangle $P Q U V$ is 2 by 7 , rectangle $W S T V$ is 3 by 5 , and rectangle $W R U V$ is 2 by 3 (since $W R=P Q=2$ and $R U=S T=3)$.

Therefore, the area of $P Q R S T V$ equals $2 \times 7+3 \times 5-2 \times 3=14+15-6=23$.