# Circle A has a center at (6 ,4 ) and a radius of 3 . Circle B has a center at (-1 ,6 ) and a radius of 5 . Do the circles overlap? If not, what is the smallest distance between them?

Jun 22, 2016

The circles overlap.

#### Explanation:

If circle A has a center at $\left(6 , 4\right)$
and circle B has a center at $\left(- 1 , 6\right)$

The distance between the centers is
$\textcolor{w h i t e}{\text{XXX}} d = \sqrt{{\left(6 - \left(- 1\right)\right)}^{2} + {\left(4 - 6\right)}^{2}}$
$\textcolor{w h i t e}{\text{XXXX}} = \sqrt{49 + 4}$
$\textcolor{w h i t e}{\text{XXXX}} = \sqrt{53}$
$\textcolor{w h i t e}{\text{XXXX}} \approx 7.28$

Along the line segment joining the centers of the two circles
A takes up 3 units
and B takes up 5 units.

Since $3 + 5 = 8$ is greater that the distance between the two centers ($7.28$) the circles overlap (by approximately $8 - 7.28 = 0.72$ units).