In the diagram, $A O B$ is a quarter circle of radius 10 and $P Q R O$ is a rectangle of perimeter 26 . The perimeter of the shaded region is
(A) $7+5 \pi$
(B) $13+5 \pi$
(C) $17+5 \pi$
(D) $7+25 \pi$
(E) $17+25 \pi$

![](https://cdn.mathpix.com/cropped/2024_04_20_e85ef69091c57cb633f3g-113.jpg?height=387&width=396&top_left_y=587&top_left_x=1274)
In order to determine the perimeter of the shaded region, we need to determine the total combined length of $\operatorname{arc} A Q B$ and segments $A P, P R$ and $R B$.

![](https://cdn.mathpix.com/cropped/2024_04_20_688a3b42f69739f2e244g-207.jpg?height=379&width=376&top_left_y=1407&top_left_x=907)

Since $A O B$ is a quarter circle of radius 10 , then $\operatorname{arc} A Q B$ has length $\frac{1}{4}(2 \pi(10))=5 \pi$.

Since $P Q R O$ is a rectangle, then $P R=Q O$ and $Q O$ is a radius of the quarter circle, so $P R=Q O=10$. So we now need to calculate $A P+R B$.

But $A P+R B=(A O-P O)+(B O-R O)=A O+B O-(P O+R O)$. We know that $A O=B O=10$ since each is a radius of the quarter circle.

Also, $P O+R O$ is half of the perimeter of the rectangle (which has total perimeter 26), so $P O+R O=13$.

Therefore, $A P+R B=10+10-13=7$.

Thus, the perimeter of the shaded region is $5 \pi+10+7=17+5 \pi$.

ANSWER: (C)