# Circle A has a center at (8 ,1 ) and an area of 72 pi. Circle B has a center at (14 ,8 ) and an area of 48 pi. Do the circles overlap?

May 17, 2016

They overlap

#### Explanation:

First circle has center at ${c}_{A} = \left(8 , 1\right)$ the second at ${c}_{B} = \left(14 , 8\right)$
their distance is $\left\lVert {c}_{A} - {c}_{B} \right\rVert = \sqrt{\left({c}_{A} - {c}_{B}\right) \left({c}_{A} - {c}_{B}\right)} = 9.21954$
Regarding areas we have ${S}_{A} = 72 \pi = \pi {r}_{A}^{2}$ and ${S}_{B} = 48 \pi = \pi {r}_{B}^{2}$ Solving for ${r}_{A} , {r}_{B}$ we have
${r}_{A} = 8.48528$ and ${r}_{B} = 6.9282$. Comparing with the center distance we have $\left\lVert {c}_{A} - {c}_{B} \right\rVert < {r}_{A} + {r}_{B}$ so they overlap