# Circle A has a center at (2 ,2 ) and a radius of 5 . Circle B has a center at (12 ,8 ) and a radius of 1 . Do the circles overlap? If not what is the smallest distance between them?

Feb 3, 2016

The circles do not overlap.
There is a minimum distance of $2 \sqrt{34} - 6 \approx 5.66$ units between them.

#### Explanation:

The length of a line segment joining the centers of the two circles is
$\textcolor{w h i t e}{\text{XXX}} \sqrt{{\left(12 - 2\right)}^{2} + {\left(8 - 2\right)}^{2}} = \sqrt{{10}^{2} + {6}^{2}} = 2 \sqrt{34}$

The distance from the center of Circle A to the edge of Circle A along the line segment joining the two circles is $5$ (the radius of A).

The distance from the edge of A to the center of Circle B along the line segment joining the centers is $2 \sqrt{34} - 5$

The distance from the center of Circle B to the edge of Circle B along the line segment joining the centers is $1$ (the radius of B).

The distance between the edges of the two circles is
$\textcolor{w h i t e}{\text{XXX}} \left(2 \sqrt{34} - 5\right) - 1$

$\textcolor{w h i t e}{\text{XXX}} = 2 \sqrt{34} - 6$

$\textcolor{w h i t e}{\text{XXX}} \approx 5.66$ (in whatever nits are being used)