# Circle A has a center at (8 ,-4 ) and a radius of 3 . Circle B has a center at (4 ,3 ) and a radius of 5 . Do the circles overlap? If not what is the smallest distance between them?

Feb 15, 2016

The circles do not overlap and are separated by a (minimum) distance of (approximately) $0.06$ units

#### Explanation:

The centers of the circles at $\left(8 , - 4\right)$ and $\left(4 , 3\right)$ are separated by a distance of
$\textcolor{w h i t e}{\text{XXX}} \sqrt{{\left(8 - 4\right)}^{2} + {\left(- 4 - 3\right)}^{2}} = \sqrt{16 + 49} = \sqrt{65} \approx 8.06$

Measured along the line segment joining the centers
$\textcolor{w h i t e}{\text{XXX}}$the radius of A uses up $3$ units
and
$\textcolor{w h i t e}{\text{XXX}}$the radius of B uses up $5$ units

which leaves
$\textcolor{w h i t e}{\text{XXX}} 8.06 - \left(3 + 5\right) = 0.06$ units along which the circles do not overlap.