# Circle A has a center at (4 ,-8 ) and a radius of 3 . Circle B has a center at (-2 ,-2 ) and a radius of 5 . Do the circles overlap? If not, what is the smallest distance between them?

Feb 2, 2016

No, since the distance of the centers is greater than the sum of the radii.

#### Explanation:

You can calculate the distance between the centers. If that distance is less than the sum of the radii, the circles do overlap. If not, they don't.

So, lets calculate the distance between $A = \left(4 , - 8\right)$ and $B \left(- 2 , - 2\right)$.
First we calculate the distance in x (called $\Delta x$) and the distance in y (called $\Delta y$).
$\Delta x = {x}_{b} - {x}_{a} = - 2 - 4 = - 6$
$\Delta y = {y}_{b} - {y}_{a} = - 2 - - 8 = - 2 + 8 = 6$

You can see this as a triangle, with base $\Delta x$ and height $\Delta y$. The distance ($d$) would then be equal to the length of the hypotenuse. We can use Pythagoras' law.
$d = \sqrt{{\left(\Delta x\right)}^{2} + {\left(\Delta y\right)}^{2}} = \sqrt{{\left(- 6\right)}^{2} + {6}^{2}} = \sqrt{36 + 36} = \sqrt{72} \approx 8.5$

This shows us that the distance between the centers of the circles is approximately $8.5$. If the circles would touch each other, the distance would be equal to $3 + 5 = 8$; the sum of the radii.
Now we have a distance greater than $8$, so the circles do not touch or overlap.