[Pythagorean Theorem (direct and inverse) $]$ [ Diameter, chords, and secants $]$

The tangent and the secant, drawn from the same point to the circle, are perpendicular to each other. The tangent is 12, and the inner part of the secant is 10. Find the radius of the circle.
The distance from the center of the circle to the secant is equal to the given tangent segment.

## Solution

Let $O$ be the center of the circle, $M$ be the common point of the tangent and the secant, $A$ be the point of tangency, $B C$ be the inner part of the secant $M C$, and $K$ be the midpoint of $B C$. Then,

$O K \perp B C$ and $O B^{2}=O K^{2}+B K^{2}=A M^{2}+B K^{2}=144+25=169$.

![](https://cdn.mathpix.com/cropped/2024_05_06_c271881b12104a3b803eg-24.jpg?height=332&width=374&top_left_y=2215&top_left_x=842)

## Answer

13.