Question

Circle A has a center at `(1 ,3 )` and an area of `16 pi`. Circle B has a center at `(2 ,7 )` and an area of `28 pi`. Do the circles overlap?

Answer 1

The distance from the centre of circle A to the centre of circle B is less than the sum of the two radii. Thus we can conclude that the two circles are overlapping.

Explanation:

area of a circle = `pi r^2`
area of circle A `= 16 pi`
area of circle B `= 28 pi`
radius of circle A = `r_A = sqrt((16pi)/pi) = sqrt(16) = 4`
radius of circle B `= r_B = sqrt((28pi)/pi) = sqrt(28) = 2sqrt(7)`

If the circles are just touching, the distance between their centres will be:
`r_A+r_B = 4 + 2sqrt(7) ~= 9.29`

The distance from the centre of circle A to the centre of circle B = d
Using Pythagorus's theorem:
`d^2 = (1-2)^2 + (3-7)^2`
`d^2 = (-1)^2 + (-4)^2`
`d^2 = (1) + (16) = 17`
`d = sqrt(17) ~= 4.12`

d is less than the sum of the two radii. Thus we can conclude that the two circles are overlapping.

They actually overlap a lot as the centre of circle A is within circle B (because `r_B > d` )!