# Circle A has a center at (3 ,5 ) and a radius of 1 . Circle B has a center at (-2 ,1 ) and a radius of 3 . Do the circles overlap? If not, what is the smallest distance between them?

Jul 27, 2016

The circles do not overlap.
There is a minimum distance of (approximately) $2.4$ between them.

#### Explanation:

The length of the line segment between A at $\left(3 , 5\right)$ and B at $\left(- 2 , 1\right)$ is given by the Pythagorean Theorem as
$\textcolor{w h i t e}{\text{XXX}} {L}_{A B} = \sqrt{{\left(3 - \left(- 2\right)\right)}^{2} + {\left(5 - 1\right)}^{2}}$
$\textcolor{w h i t e}{\text{XXXXXX}} = \sqrt{41}$
$\textcolor{w h i t e}{\text{XXXXXX}} \approx 6.4$

The radius of A covers $1$ unit of this length
and the radius of B covers another $3$ units
for a total of $4$ units covered by one or the other of the circles.

This leaves $6.4 - 4 = 2.4$ units uncovered
and this is the minimum distance between the two circles.