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From one point, two tangents are drawn to a circle. The length of each tangent is 12, and the distance between the points of tangency is 14.4. Find the radius of the circle.

#
The angle between the given chord and the radius drawn to the point of tangency is equal to half the angle between the tangents.

## Solution

Let $A$ be the given point, $B$ and $C$ be the points of tangency, and $O$ be the center of the circle. Since the line $O A$ is perpendicular to the segment $B C$ and passes through its midpoint $M$, then

$$
\angle O C B = \angle O A C, \sin \angle O A C = \frac{C M}{A C} = \frac{7.2}{12} = \frac{3}{5}
$$

Therefore, $\cos \angle O A C = \frac{4}{5}$. From triangle $OCM$, we find that

$$
O C = \frac{C M}{\cos \angle O C M} = 9
$$

![](https://cdn.mathpix.com/cropped/2024_05_06_d1936174aeda27dcc10dg-16.jpg?height=454&width=458&top_left_y=1808&top_left_x=800)

## Answer

9.