## Task B-1.3.

Tangents drawn from point $B$ to a circle of radius $r$ touch the circle at points $A$ and $C$. Determine the area of the figure bounded by the tangents $B C, B A$ and the smaller circular arc $\widetilde{A C}$, if the measure of angle $\varangle A B C$ is $60^{\circ}$.
## Solution.

![](https://cdn.mathpix.com/cropped/2024_05_30_964e2f56f6f4ff54223dg-02.jpg?height=512&width=625&top_left_y=535&top_left_x=641)

The desired area is equal to the area of the quadrilateral $A B C S$ minus the area of the circular sector. The area of this quadrilateral is equal to twice the area of the right triangle $A B S$, which is half of an equilateral triangle (its angles are $30^{\circ}, 60^{\circ}$, and $90^{\circ}$).

From this, the lengths of the sides of triangle $A B S$ are $|A S|=r, |A B|=\frac{2 r \sqrt{3}}{2}=r \sqrt{3}$.

$2 \cdot P_{\triangle A B S}=2 \cdot \frac{r \cdot r \sqrt{3}}{2}=r^{2} \sqrt{3}$.

1 point

The angle $\varangle A B C=60^{\circ}$, so the central angle of the circular sector is $\varangle A S C=120^{\circ}$.

The area of the circular sector with a central angle of $120^{\circ}$ is one-third of the area of the given circle, i.e., $P_{i}=\frac{1}{3} r^{2} \pi$.

The desired area is $P=r^{2} \sqrt{3}-\frac{1}{3} r^{2} \pi$.