The minimum distance between points of two concentric circles is 2, and the maximum distance is 16. Find the radii of the circles.

#
The smallest distance between points of two concentric circles is the difference of their radii, and the largest is the sum.

## Solution

Let $O$ be the center of the circles, $r$ and $R$ be their radii ($r < R$). If $X$ and $Y$ are arbitrary points on the smaller and larger circles, respectively, then

$$
\begin{gathered}
r + XY = OX + XY \geqslant OY, \text{ or } XY \geqslant R - r \\
XY \leqslant OX + OY = r + R
\end{gathered}
$$

Therefore, $R + r = 16$ and $R - r = 2$. From the resulting system of equations, we find that $R = 9$ and $r = 7$.

![](https://cdn.mathpix.com/cropped/2024_05_06_16ab69fab1743849dba9g-14.jpg?height=371&width=397&top_left_y=752&top_left_x=839)

Answer

7 and 9.