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?

Feb 8, 2016

Yes, they overlap.

Explanation:

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 $\le 9$ from the center of A are inside or on A. Similarly, all points with distances $\le 5$ 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 $\sqrt{{\left(1 - \left(- 6\right)\right)}^{2} + {\left(- 5 - 4\right)}^{2}} = \sqrt{49 + 81} = \sqrt{130} \approx 11.4$

As $11.4 < 14$, the circles overlap.