[ Product of the lengths of the chord segments and the lengths of the secant segments ]

A line passing through point $A$, located outside the circle at a distance of 7 from its center, intersects the circle at points $B$ and $C$. Find the radius of the circle, given that $AB=3$, $BC=5$.

#
The product of the entire secant and its external part for a given point and a given circle is constant.

## Solution

Since $A B<B C$, point $B$ lies between points $A$ and $C$. Let $O$ be the center of the circle, $R$ its radius, and the line $A O$ intersects the circle at points $D$ and $E$ (with $D$ between $A$ and $O$). Then

$$
A D \cdot A E=A B \cdot A C, (7-R)(7+R)=3 \cdot 8, 49-R^{2}=24
$$

$R^{2}=25$. Therefore, $R=5$.

## Answer

5.