# Circle A has a center at (3 ,7 ) and a radius of 2 . Circle B has a center at (1 ,3 ) and a radius of 4 . Do the circles overlap? If not, what is the smallest distance between them?

Dec 23, 2016

The circles overlap.

#### Explanation:

The distance between the center of A at $\left(3 , 7\right)$ and B at $\left(1 , 3\right)$ is given by the Pythagorean Theorem as
$\textcolor{w h i t e}{\text{XXX}} d = \sqrt{{\left(3 - 1\right)}^{2} + {\left(7 - 3\right)}^{2}} = \sqrt{{2}^{2} + {4}^{2}} = \sqrt{20} \approx 4.47$

Circle A with a radius of $2$ covers $2$ units of the distance between the centers of A and B.
Circle B with a radius of $4$ covers $4$ units of the distance between the centers of A and B.

Together the circles cover $2 + 4 = 6$ which is more than the distance between the centers;
so the circles overlap (by approximately $6 - 4.47 = 1.53$ units).