# Circle A has a center at (8 ,4 ) and a radius of 3 . Circle B has a center at (-2 ,-2 ) and a radius of 7 . Do the circles overlap? If not, what is the smallest distance between them?

Feb 10, 2016

The circles do not overlap; they are separated by a distance of
$\sqrt{136} - 10 \approx 1.66$

#### Explanation:

Distance between $\left(8 , 4\right)$ and $\left(- 2 , - 2\right)$ is
$\textcolor{w h i t e}{\text{XXX}} \sqrt{{\left(8 - \left(- 2\right)\right)}^{2} + {\left(4 - \left(- 2\right)\right)}^{2}} = \sqrt{{10}^{2} + {6}^{2}} = \sqrt{136}$

If Circle A has a radius $3$ and Circle B has a radius $7$

then along the line segment joining the centers of the two circles
the circles are separated by a distance of
$\textcolor{w h i t e}{\text{XXX}} \sqrt{136} - \left(3 + 7\right) = \sqrt{136} - 10 \approx 1.66$