# Circle A has a center at (8 ,4 ) and a radius of 3 . Circle B has a center at (-1 ,-2 ) and a radius of 4 . Do the circles overlap? If not, what is the smallest distance between them?

Feb 14, 2017

The circles are separate and there is a minimum distance between them of 3.82

#### Explanation:

The distance between the two centres can be calculated using this
formula which is derived from Pythagoras's theorem
$\sqrt{{\left({x}_{2} - {x}_{1}\right)}^{2} + {\left({y}_{2} - {y}_{1}\right)}^{2}}$

substituting in the values from the question
$\sqrt{{\left(8 - - 1\right)}^{2} + {\left(4 - - 2\right)}^{2}}$

Simplifying
$\sqrt{81 + 36} = 10.82$

subtracting the radii of the two circles from the distance between the two centres indicates whether they over lap, touch or are separate.

distance - radius A - radius B < 0 : the circles overlap
distance - radius A - radius B = 0 : the circles touch
distance - radius A - radius B > 0 : the circles are separate

In this case
$10.82 - 3 - 4 = 3.82$ (greater than 0)
The circles are separate and there is a minimum distance between them of 3.82