Circle A has a center at #(-6 ,4 )# and a radius of #9 #. Circle B has a center at #(1 ,-5 )# and a radius of #5 #. Do the circles overlap? If not, what is the smallest distance between them?

Question

Circle A has a center at #(-6 ,4   )# and a radius of #9    #.  Circle B has a center at #(1  ,-5   )# and a radius of #5 #.  Do the circles overlap?  If not, what is the smallest distance between them?

Avery Allen · Accepted Answer

Yes, they overlap.

As the radii of the circles sum to #14#, the centers of circle A and circle B need to be at greater than #14# units apart for the circles to not overlap at any point. To see why this is the case, construct the line segment from the center of A to the center of B. All points on the segment with distance #<=9# from the center of A are inside or on A. Similarly, all points with distances #<=5# from the center of B are inside or on B. For these two sections of the segment to not intersect, the line segment would need to be greater than #14# units in length. The distance between the center of circle A and the center of circle B is #sqrt((1-(-6))^2+(-5-4)^2)=sqrt(49+81)=sqrt(130)~~11.4# As #11.4 < 14#, the circles overlap.

Sophia Alfred · Answer

undefined The circles do overlap. The distance between the centers of the circles is $ \sqrt{(1 - (-6))^2 + (-5 - 4)^2} = \sqrt{7^2 + (-9)^2} = \sqrt{49 + 81} = \sqrt{130} \approx 11.4 $. Since this distance is less than the sum of the radii of the circles (9 + 5 = 14), the circles overlap.