# Circle A has a center at (7 ,-5 ) and a radius of 1 . Circle B has a center at (4 ,2 ) and a radius of 4 . Do the circles overlap? If not, what is the smallest distance between them?

Feb 2, 2016

The circles don't overlap, and the smallest distance between them = 2.62

#### Explanation:

• First: using the formula :
$d = \sqrt{{\left({x}_{1} - {x}_{2}\right)}^{2} + {\left({y}_{1} - {y}_{2}\right)}^{2}}$

the distance between the two centers$= d = \sqrt{{\left(7 - 4\right)}^{2} + {\left(- 5 - 2\right)}^{2}} = \sqrt{9 + 49}$
$= \sqrt{58} = 7.62$

• Second:
The distance between the two centers > the sum of the two radii.
Meaning the circles don't overlap.

• Third: The smallest distance between them is the portion of the line connecting the two centers that doesn't fall in either circle. Or $s$ in the image below.

$s = \overline{P O} - \text{sum of the two radii}$
$s = \sqrt{58} - 5$
$s = 2.62$