In the diagram, square $P Q R S$ is $3 \times 3$. Points $T$ and $U$ are on side $Q R$ with $Q T=T U=U R=1$. Points $V$ and $W$ are on side $R S$ with $R V=V W=W S=1$. Line segments $T X$ and $U Y$ are perpendicular to $Q R$ and line segments $V Y$ and $W X$ are perpendicular to $R S$. The ratio of the shaded area to the unshaded area is
(A) $2: 1$
(B) $7: 3$
(C) $7: 4$
(D) $5: 4$
(E) $3: 1$

![](https://cdn.mathpix.com/cropped/2024_04_20_e85ef69091c57cb633f3g-051.jpg?height=331&width=352&top_left_y=973&top_left_x=1293)

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

Since $P Q R S$ is a square and $T X$ and $U Y$ are perpendicular to $Q R$, then

$T X$ and $U Y$ are parallel to $P Q$ and $S R$.

Similarly, $V Y$ and $W X$ are parallel to $P S$ and $Q R$.

Therefore, if we extend $W X$ and $V Y$ to meet $P Q$ and extend $T X$ and

$U Y$ to meet $P S$, then square $P Q R S$ is divided into 9 rectangles.

Since $Q T=T U=U R=1$ and $R V=V W=W S=1$, then in fact

$P Q R S$ is divided into 9 squares, each of which is 1 by 1 .

Of these 9 squares, 6 are shaded and 3 are unshaded.

![](https://cdn.mathpix.com/cropped/2024_04_20_688a3b42f69739f2e244g-101.jpg?height=325&width=371&top_left_y=1014&top_left_x=1582)

Therefore, the ratio of the shaded area to the unshaded area is $6: 3$, which equals $2: 1$.

## Solution 2

Consider quadrilateral $Y U R V$.

$Y U R V$ has three right angles: at $U$ and $V$ because $U Y$ and $V Y$ are perpendicular to $Q R$ and $R S$, respectively, and at $R$ because $P Q R S$ is a square. Since $Y U R V$ has three right angles, then it has four right angles and so is a rectangle.

Since $R V=U R=1$, then $Y U R V$ is actually a square and has side length 1 , and so has area $1^{2}$, or 1 .

Similarly, $X T R W$ is a square of side length 2 , and so has area $2^{2}$, or 4 .

![](https://cdn.mathpix.com/cropped/2024_04_20_688a3b42f69739f2e244g-101.jpg?height=328&width=371&top_left_y=1560&top_left_x=1582)

Since square $P Q R S$ is $3 \times 3$, then its area is $3^{2}$, or 9 .

The area of the unshaded region is equal to the difference between the areas of square $X T R W$ and square $Y U R V$, or $4-1=3$.

Since square $P Q R S$ has area 9 and the area of the unshaded region is 3 , then the area of the shaded region is $9-3=6$.

Finally, the ratio of the shaded area to the unshaded area is $6: 3$, which equals $2: 1$.

ANSWER: (A)