[Theorem on the lengths of a tangent and a secant; the product of the entire secant and its external part. Pythagorean Theorem (direct and inverse).

From a point $A$ outside a circle, a tangent and a secant are drawn to the circle. The distance from point $A$ to the point of tangency is 16, and the distance from point $A$ to one of the points of intersection of the secant with the circle is 32. Find the radius of the circle if the distance from its center to the secant is 5.

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Let a secant intersect a circle at points $B$ and $C$, and $M$ be the point of tangency. Then $A M=16, A C=32, B C=32$. By the tangent-secant theorem, $A M^{2}=A C \cdot A B$, or $16^{2}=32(32-B C)$. From this, $B C=24$.

Let $K$ be the projection of the center $O$ of the given circle onto the chord $B C$. The radius of the circle is found using the Pythagorean theorem: $R^{2}=O B^{2}=O K^{2}+B K^{2}=169$.

![](https://cdn.mathpix.com/cropped/2024_05_06_36c4956010bd91c4b4eeg-07.jpg?height=349&width=443&top_left_y=2421&top_left_x=813)

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