# Circle A has a center at (1 ,5 ) and an area of 18 pi. Circle B has a center at (8 ,4 ) and an area of 66 pi. Do the circles overlap?

Jun 5, 2016

Get the distance between the centers.
For the circles to overlap, the distance between the centers should be less than or equal to the sum of the radii but greater than or equal to the difference

${A}_{A} = \pi {r}_{A}^{2} = 18 \pi$

$\implies {r}_{A}^{2} = 18$

$\implies {r}_{A} = 3 \sqrt{2} \approx 4.2 \ldots$

${A}_{B} = \pi {r}_{B}^{2} = 66 \pi$

$\implies {r}_{B}^{2} = 66$

$\implies {r}_{B} = \sqrt{66} \approx 8. . .$

Distance between centers:

$D = \sqrt{{\left({x}_{A} - {x}_{B}\right)}^{2} + {\left({y}_{A} - {y}_{B}\right)}^{2}}$

$\implies D = \sqrt{{\left(1 - 8\right)}^{2} + {\left(5 - 4\right)}^{2}}$

$\implies D = \sqrt{{\left(- 7\right)}^{2} + {1}^{2}}$

$\implies D = \sqrt{50}$

$\implies D = 5 \sqrt{2} \approx 7. . .$

$| {r}_{A} - {r}_{B} | \approx 3.8$

${r}_{A} + {r}_{B} \approx 12.2$

The approximate distance between the centers is less than the sum of the radii but greater than the difference, the circles should overlap.