8,9

Two tangents are drawn to a circle of radius 7 from a point located at a distance of 25 from the center.

Find the distance between the points of tangency.
Let $M A$ and $M B$ be the given tangents, $O$ be the center of the circle, and $K$ be the midpoint of $A B$. Then $M A^{2}=O M^{2}-O A^{2}=24^{2}$. Since $O M \cdot A K=2 S_{O A M}=A M \cdot A O$, then $A B=2 A K=2 \cdot \frac{A M}{O M} \cdot A O=13.44$.

## Answer

$13.44$.

Two circles with radii $r$ and $R$ touch each other externally. A tangent is drawn from the center of one circle to the other, and from the point of tangency, a tangent is drawn to the first circle. Find the length of the last tangent.

## Hint

The line connecting the centers of two touching circles passes through their point of tangency.

## Solution

Let $O$ and $O_{2}$ be the centers of the circles, $M$ and $K$ be the first and second points of tangency (see figure).

By the Pythagorean theorem, $O M^{2}=O_{2} O^{2}-O_{2} M^{2}=(R+r)^{2}-R^{2}=r^{2}+2 r R, M K^{2}=O M^{2}-O K^{2}=2 r R$.

![](https://cdn.mathpix.com/cropped/2024_05_06_14311972f49418f31424g-13.jpg?height=383&width=594&top_left_y=706&top_left_x=732)

Answer

$\sqrt{2 r R}$.