# Circle A has a center at (5 ,-4 ) and a radius of 6 . Circle B has a center at (-4 ,-8 ) and a radius of 5 . Do the circles overlap? If not, what is the smallest distance between them?

Sep 20, 2016

The circles overlap.

#### Explanation:

The distance between the center of A at color(blue)(""(5,-4))
and the center of B at color(green)(""(-4,-8))
is
$\textcolor{w h i t e}{\text{XXX}} d = \sqrt{{\left(\textcolor{b l u e}{5} - \left(\textcolor{g r e e n}{- 4}\right)\right)}^{2} + {\left(\left(\textcolor{b l u e}{- 4}\right) - \left(\textcolor{g r e e n}{- 8}\right)\right)}^{2}}$

$\textcolor{w h i t e}{\text{XXX}} = \sqrt{{9}^{2} + {4}^{2}}$

$\textcolor{w h i t e}{\text{XXX}} = \sqrt{97}$

$\textcolor{w h i t e}{\text{XXX}} \approx 9.848858$

Circle A with a radius of $\textcolor{red}{6}$
covers $\textcolor{red}{6}$ units of this distance
and
Circle B with a radius of $\textcolor{m a \ge n t a}{5}$
covers $\textcolor{m a \ge n t a}{5}$ units of this distance

Between the two circles,
they cover $\textcolor{red}{6} + \textcolor{m a \ge n t a}{5} = 11$ units.

Since the distance between them is only (approx.) $9.85$
the circles must overlap ( by approx. $11 - 9.85 = 1.15$

As verification, here is what the circles look like: 