[ Triangle Inequality ]

The radius of the circle is 10, and a given point is 15 units away from the center. Find the smallest and largest distances from this point to the points on the circle.

#
To find the smallest and largest distances from a given point to the points of a circle, use the triangle inequality.

## Solution

Let $M$ be the given point, $O$ the center of the circle, $AB$ its diameter lying on the line $OM$ (with $A$ between $O$ and $M$), and $X$ any point on the circle. Then

$$
M X + O X \geqslant O M = O A + A M
$$

Therefore,

$$
M X \geqslant A M = 15 - 10 = 5
$$

Similarly, we find that $M X \leqslant 15 + 10 = 25$.

![](https://cdn.mathpix.com/cropped/2024_05_06_b0b755a355dbd05a96cdg-30.jpg?height=303&width=644&top_left_y=723&top_left_x=707)

## Answer

5 and 25.