# Circle A has a center at (8 ,7 ) and an area of 81 pi. Circle B has a center at (4 ,2 ) and an area of 36 pi. Do the circles overlap? If not, what is the shortest distance between them?

Feb 16, 2016

The circles overlap.

#### Explanation:

The distance between the centers of Circle A and Circle B is
$\textcolor{w h i t e}{\text{XXX}} d = \sqrt{{\left(8 - 4\right)}^{2} + {\left(7 - 2\right)}^{2}} = \sqrt{41} \approx 6.403$

${\text{Area}}_{A} = \pi {r}_{A}^{2} = 81 \pi$
$\textcolor{w h i t e}{\text{XXX}} \rightarrow {r}_{A}^{2} = 81$
$\textcolor{w h i t e}{\text{XXX}} \rightarrow {r}_{a} = 9$

${\text{Area}}_{B} = \pi {r}_{B}^{2} = 36 \pi$
$\textcolor{w h i t e}{\text{XXX}} \rightarrow {r}_{B}^{2} = 36$
$\textcolor{w h i t e}{\text{XXX}} \rightarrow {r}_{B} = 6$

Since the sum of the radii is greater than the distance between the centers, the circles overlap.

In fact along the line joining the centers of the circles, the circles overlap by
color(white)("XXX")(9+6) - 6.403 = 8.597 " units"