The radius of the circle is 10, and a given point is located at a distance of 3 from its center. Find the smallest and largest distances from this point to the points on the circle.

#
The smallest and largest distances from point $M$ to the points on the circle are equal to the lengths of the segments into which point $M$ divides the diameter passing through it.

## Solution

Let $M$ be the given point, $O$ the center of the circle, $AB$ the diameter passing through point $M$ ($M$ between $O$ and $A$), and $X$ any point on the circle. Then

$$
\begin{gathered}
O X \leqslant O M + M X, \quad M X \geqslant O X - O M = O A - O M = 10 - 3 = 7 \\
M X \leqslant O X + O M = O B + O M = 10 + 3 = 13
\end{gathered}
$$

![](https://cdn.mathpix.com/cropped/2024_05_06_76207e7feb35677daaaeg-48.jpg?height=248&width=354&top_left_y=46&top_left_x=869)

## Answer

7 and 13.