# Circle A has a center at (2 ,3 ) and an area of 8 pi. Circle B has a center at (13 ,7 ) and an area of 54 pi. Do the circles overlap?

Mar 24, 2016

This straightforward application of distance formula and comparing of the radiuses.
$R = 10.18 < {D}_{\left(B - A\right)} = 11.7$ so no overlap

#### Explanation:

This straightforward application of distance formula and comparing of the radiuses. First let calculate the radii:
Let A(2,3) and B(13,7)
${r}_{A} = \pi {r}^{2} = 8 \pi \implies r = 2 \sqrt{2}$
${r}_{B} = \pi {r}^{2} = 54 \pi \implies r = 3 \sqrt{6}$
so the sum of the radii, $R = {r}_{B} + {r}_{A} = 2 \sqrt{2} + 3 \sqrt{6}$
$R = \sqrt{2} \left(2 + 3 \sqrt{3}\right)$

Now use the distance formula to find the distance from the center of the circles:
${D}_{\left(B - A\right)} = \sqrt{{\left(13 - 2\right)}^{2} + {\left(7 - 3\right)}^{2}} = \sqrt{137}$

Now $R = 10.18 < {D}_{\left(B - A\right)} = 11.7$ so no overlap