| Two tangents drawn from one point |  |
| :---: | :---: | :---: |
|  | Properties and characteristics of tangents | Slo |
|  | Auxiliary similar triangles  Pythagoras' theorem (direct and inverse) |  |

Two tangents are drawn from one point to a circle. The length of the tangent is 156, and the distance between the points of tangency is 120. Find the radius of the circle.
Let $O$ be the center of the circle, $M A$ and $M B$ be the tangents, $A$ and $B$ be the points of tangency, and $K$ be the midpoint of segment $A B$. Then $M K^{2} = A M^{2} - A K^{2} = 156^{2} - 60^{2} = 96 \cdot 216 = 144^{2}$.

From the similarity of triangles $M A O$ and $M K A$, it follows that $O A : A M = A K : M K$. Therefore, $O A = A M^{A K} / M R = 65$.

![](https://cdn.mathpix.com/cropped/2024_05_06_d3cd76ad0dc53ae543bdg-30.jpg?height=274&width=472&top_left_y=1512&top_left_x=793)

## Answer

65.