Question

Circle A has a center at `(3 ,7 )` and a radius of `2`. Circle B has a center at `(1 ,3 )` and a radius of `4`. Do the circles overlap? If not, what is the smallest distance between them?

Answer 1

The circles overlap.

Explanation:

The distance between the center of A at `(3,7)` and B at `(1,3)` is given by the Pythagorean Theorem as
`color(white)("XXX")d=sqrt((3-1)^2+(7-3)^2)=sqrt(2^2+4^2)=sqrt(20)~~4.47`

Circle A with a radius of `2` covers `2` units of the distance between the centers of A and B.
Circle B with a radius of `4` covers `4` units of the distance between the centers of A and B.

Together the circles cover `2+4=6` which is more than the distance between the centers;
so the circles overlap (by approximately `6-4.47=1.53` units).