In the diagram shown,

- $S T U V$ is a square,
- $Q$ and $P$ are the midpoints of $S T$ and $U V$,
- $P R=Q R$, and
- $V Q$ is parallel to $P R$.

What is the ratio of the shaded area to the unshaded

![](https://cdn.mathpix.com/cropped/2024_04_20_46ead6524a8d61e21c51g-042.jpg?height=360&width=345&top_left_y=1441&top_left_x=1386)
area?
(A) $2: 3$
(B) $3: 5$
(C) $1: 1$
(D) $7: 9$
(E) $5: 7$
We begin by joining $Q$ to $P$.

Since $Q$ and $P$ are the midpoints of $S T$ and $U V$, then $Q P$ is parallel to both $S V$ and $T U$ and rectangles $S Q P V$ and $Q T U P$ are identical.

In rectangle $S Q P V, V Q$ is a diagonal.

Similarly, since $P R$ is parallel to $V Q$ then $P R$ extended to $T$ is a diagonal of rectangle $Q T U P$, as shown in Figure 1.

In Figure 2, we label points $A, B, C, D, E$, and $F$, the midpoints of $S Q, Q T, T U, U P, P V$, and $V S$, respectively.

We join $A$ to $E, B$ to $D$ and $F$ to $C$, with $F C$ intersecting $Q P$ at the centre of the square $O$, as shown.

Since $P R=Q R$ and $R$ lies on diagonal $P T$, then both $F C$ and $B D$ pass through $R$. (That is, $R$ is the centre of $Q T U P$.)

The line segments $A E, Q P, B D$, and $F C$ divide square $S T U V$ into 8 identical rectangles.

In one of these rectangles, $Q B R O$, diagonal $Q R$ divides the rectangle into 2 equal areas.

That is, the area of $\triangle Q O R$ is half of the area of rectangle $Q B R O$.

![](https://cdn.mathpix.com/cropped/2024_04_20_748fdafcde2a47412608g-192.jpg?height=366&width=336&top_left_y=191&top_left_x=1534)

Figure 1

![](https://cdn.mathpix.com/cropped/2024_04_20_748fdafcde2a47412608g-192.jpg?height=364&width=371&top_left_y=669&top_left_x=1495)

Figure 2

Similarly, the area of $\triangle P O R$ is half of the area of rectangle

PORD.

Rectangle $S Q P V$ has area equal to 4 of the 8 identical rectangles.

Therefore, $\triangle Q P V$ has area equal to 2 of the 8 identical rectangles (since diagonal $V Q$ divides the area of $S Q P V$ in half).

Thus the total shaded area, which is $\triangle Q O R+\triangle P O R+\triangle Q P V$, is equivalent to the area of $\frac{1}{2}+\frac{1}{2}+2$ or 3 of the identical rectangles.

Since square $S T U V$ is divided into 8 of these identical rectangles, and the shaded area is equivalent to the area of 3 of these 8 rectangles, then the unshaded area occupies an area equal to that of the remaining $8-3$ or 5 rectangles.

Therefore, the ratio of the shaded area to the unshaded area is $3: 5$.

ANSWER: (B)