In the diagram, $O$ is the centre of a circle with radii $O P=O Q=5$. The perimeter of the shaded region, including the two radii, is closest to
(A) 34
(B) 41
(C) 52
(D) 59
(E) 68

![](https://cdn.mathpix.com/cropped/2024_04_20_6ed09463f225f8ba1f07g-087.jpg?height=312&width=301&top_left_y=1584&top_left_x=1302)
The perimeter of the shaded region equals the sum of the lengths of $O P$ and $O Q$ plus the length of $\operatorname{arc} P Q$.

Each of $O P$ and $O Q$ has length 5 .

Arc $P Q$ forms $\frac{3}{4}$ of the circle with centre $O$ and radius 5, because the missing portion corresponds to a central angle of $90^{\circ}$, and so is $\frac{1}{4}$ of the total circle.

Thus, the length of $\operatorname{arc} P Q$ is $\frac{3}{4}$ of the circumference of this circle, or $\frac{3}{4}(2 \pi(5))=\frac{15}{2} \pi$.

Therefore, the perimeter is $5+5+\frac{15}{2} \pi \approx 33.56$ which, of the given answers, is closest to 34 .

ANSWER: (A)