$[$ Theorem on the lengths of a tangent and a secant; the product of the entire secant and its external part

A point $M$ is connected to a circle by two lines. One of them touches the circle at point $A$, while the other intersects the circle at points $B$ and $C$, with $B C=7$ and $B M=9$. Find $A M$.

#
Use the Tangent-Secant Theorem. Consider two cases.

## Solution

Let point $B$ lie between points $M$ and $C$. According to the Tangent-Secant Theorem,

$$
A M^{2}=M C \cdot M B=(9+7) 9=16 \cdot 9=12^{2}
$$

Therefore, $A M=12$.

If point $C$ lies between points $B$ and $M$, then similarly we get $A M=3 \sqrt{2}$.

## Answer

12 or $3 \sqrt{2}$.