Circle A has a center at (5 ,-2 ) and a radius of 2 . Circle B has a center at (4 ,6 ) and a radius of 3 . Do the circles overlap? If not what is the smallest distance between them?

Jun 18, 2016

They do not overlap; there is a minimum distance between them of $3.06$ units (approximately)

Explanation:

Given circles with centers $\left(5 , - 2\right)$ and $\left(4 , 6\right)$
the distance between the centers can be calculated using the Pythagorean Theorem:
$\textcolor{w h i t e}{\text{XXX}} d = \sqrt{{\left(4 - 5\right)}^{2} + {\left(6 - \left(- 2\right)\right)}^{2}} = \sqrt{{\left(- 1\right)}^{2} + {8}^{2}} = \sqrt{65}$

Using a calculator $d \approx 8.06$

Measuring along a line joining the two centers:
one circle's radius takes up 2 units and
the other circle's radius takes up 3 units.

Since the sum of the two radii do not make up the distance between the centers, the circles do not overlap,
and the minimum distance between them is (approximately)
$\textcolor{w h i t e}{\text{XXX}} 8.06 - \left(2 + 3\right) = 3.06$