# Circle A has a center at (12 ,9 ) and an area of 25 pi. Circle B has a center at (3 ,1 ) and an area of 64 pi. Do the circles overlap?

Mar 17, 2018

Yes

#### Explanation:

First we must find the distance between the centers of the two circles. This is because this distance is where the circles will be closest together, so if they overlap it will be along this line. To find this distance we can use the distance formula: $d = \sqrt{{\left({x}_{1} - {x}_{2}\right)}^{2} + {\left({y}_{1} - {y}_{2}\right)}^{2}}$

$d = \sqrt{{\left(12 - 3\right)}^{2} + {\left(9 - 1\right)}^{2}} = \sqrt{81 + 64} = \sqrt{145} \approx 12.04$

Now we must find the radius of each circle. We know the area of a circle is $\pi {r}^{2}$, so we can use that to solve for r.

$\pi {\left({r}_{1}\right)}^{2} = 25 \pi$
${\left({r}_{1}\right)}^{2} = 25$
${r}_{1} = 5$

$\pi {\left({r}_{2}\right)}^{2} = 64 \pi$
${\left({r}_{2}\right)}^{2} = 64$
${r}_{2} = 8$

Finally we add these two radii together. The sum of the radii is 13, which is greater than the distance between the centers of the circle, meaning that the circles will overlap.