In the diagram, the circle has centre $O$ and radius 6 . Point $A$ is outside the circle and points $B$ and $C$ are on the circle so that $A B$ is perpendicular to $B O, A C$ is perpendicular to $C O$, and $\angle B A C=50^{\circ}$. What is the area of the shaded region?

![](https://cdn.mathpix.com/cropped/2024_04_17_2efeb2001664c4e9869eg-2.jpg?height=396&width=572&top_left_y=325&top_left_x=836)
Since $A B O C$ is a quadrilateral, then the sum of its interior angles is $360^{\circ}$.

Thus, $\angle B O C=360^{\circ}-\angle B A C-\angle A B O-\angle A C O=360^{\circ}-50^{\circ}-90^{\circ}-90^{\circ}=130^{\circ}$.

Therefore, the shaded region is a sector of the circle with central angle $130^{\circ}$.

Since the radius of the circle is 6 , then the area of the entire circle is $\pi 6^{2}=36 \pi$, so the area of the shaded sector is $\frac{130^{\circ}}{360^{\circ}}(36 \pi)=\frac{13}{36}(36 \pi)=13 \pi$.

ANSWER: $13 \pi$