Question

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?

Answer 1

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 = pir_A^2 = 18pi`

`=> r_A^2 = 18`

`=> r_A = 3sqrt2 ~~ 4.2...`

`A_B = pir_B^2 = 66pi`

`=> r_B^2 = 66`

`=> r_B = sqrt66 ~~ 8...`

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Distance between centers:

`D = sqrt((x_A - x_B)^2 + (y_A - y_B)^2)`

`=> D = sqrt((1 - 8)^2 + (5 - 4)^2)`

`=> D = sqrt((-7)^2 + 1^2)`

`=> D = sqrt50`

`=> D = 5sqrt2 ~~ 7...`

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`|r_A - r_B| ~~ 3.8`

`r_A + r_B ~~ 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.