Question

Circle A has a center at `(3 ,2 )` and an area of `96 pi`. Circle B has a center at `(12 ,7 )` and an area of `48 pi`. Do the circles overlap?

Answer 1

Both circles overlap.

Explanation:

Circle `A`, Area`=pir^2=96pi`
Solving for radius `r`,
`pir^2=96pi`
or `r^2=96`
or `r=sqrt96`
`r_A=4sqrt6`
`r_A=9.8`, rounded to one decimal place
Similarly Circle `B`
Area`=pir^2=48pi`
Solving for radius `r`,
`pir^2=48pi`
or `r^2=48`
or `r=sqrt48`
`r_B=4sqrt3`
`r_B=4sqrt3`
`r_B=6.4`, rounded to one decimal place
Now `r_A+r_B=9.8+6.4=16.2`, rounded to one decimal place

Distance between the two centers, `(3,2) and (12,7)`
`d=sqrt((x_2-x_1)^2+(y_2-y_1)^2)`
or `d=sqrt((12-3)^2+(7-2)^2)`
or `d=sqrt((9)^2+(5)^2)`
or `d=sqrt 106`
or `d=10.3`, rounded to one decimal place

Since the sum of both the radii is `16.2>d`, the distance between the two centers.
Therefore, both the circles overlap.