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?

1 Answer
Dec 23, 2016

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).