jmc

The perimeter of the shaded region equals the sum of the lengths of $OP$ and $OQ$ plus the length of arc $PQ$.

Each of $OP$ and $OQ$ has length 5.

Arc $PQ$ forms $\frac{3}{4}$ of the circle with center $O$ 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 arc $PQ$ 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 $5+5+\frac{15}{2}\pi =\boxed{10+\frac{15}{2}\pi }$.