# Circle A has a center at (9 ,-2 ) and a radius of 1 . Circle B has a center at (1 ,3 ) and a radius of 3 . Do the circles overlap? If not, what is the smallest distance between them?

Nov 4, 2016

The circles do not overlap. There is a distance of (aprrox.) $5.434$ between them.

#### Explanation:

The distance from the center of A at $\left(9 , - 2\right)$ and the center of B at $\left(1 , 3\right)$ is
$\textcolor{w h i t e}{\text{XXX}} d = \sqrt{{\left(9 - 1\right)}^{2} + {\left(- 2 - 3\right)}^{2}} = \sqrt{64 + 25} = \sqrt{89}$

$\textcolor{w h i t e}{\text{XXXx}} \approx 9.434$ (using a calculator)

Circle A with a radius of $1$ covers $1$ unit of this distance.
Circle B with a radius of $3$ covers $3$ units of this distance.

This leaves a minimum of $9.434 - \left(1 + 3\right) = 5.434$ units uncovered between the two circles.