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

Feb 7, 2016

For the circles to overlap, the distance between the centers should be less than or equal to the sum of the circles' respective radius. If the circles, do not overlap, then the smallest distance between the two would be the difference between the distance between the centers and the sum of their radii.

$d = \sqrt{{\left({x}_{1} - {x}_{2}\right)}^{2} + {\left({y}_{1} - {y}_{2}\right)}^{2}}$

$\implies d = \sqrt{{\left(2 - 7\right)}^{2} + {\left(7 - 2\right)}^{2}}$

$\implies d = \sqrt{{5}^{2} + {5}^{2}}$

$\implies d = \sqrt{2 \left({5}^{2}\right)}$

$\implies d = 5 \sqrt{2}$

$\implies d \approx 7$

Since the distance between the centers is greater than the sum of the radii (which is 6), the circles do not overlap.

The smallest distance between them is

$d ' \approx 7 - 6 \approx 1$.

If you need the precise value of the distance between the circles, go get a calculator and get the value of $\sqrt{2}$ (I used $\sqrt{2} \approx 1.4$).