# Circle A has a center at (3 ,2 ) and a radius of 6 . Circle B has a center at (-2 ,1 ) and a radius of 3 . Do the circles overlap? If not, what is the smallest distance between them?

Mar 14, 2016

The distance $d \left(A , B\right)$ and the radius of each circle ${r}_{A}$ and ${r}_{B}$ must satisfy the condition:

$d \left(A , B\right) \le {r}_{A} + {r}_{B}$

In this case, they do, so the circles overlap.

#### Explanation:

If the two circles overlap, this means that the least distance $d \left(A , B\right)$ between their centers must be less than the sum of their radius, as it can be understood from the picture:

(numbers in picture are random from the internet)

So to overlap at least once:

$d \left(A , B\right) \le {r}_{A} + {r}_{B}$

The Euclidean distance $d \left(A , B\right)$ can be calculated:

d(A,B)=sqrt((Δx)^2+(Δy)^2)

Therefore:

$d \left(A , B\right) \le {r}_{A} + {r}_{B}$

sqrt((Δx)^2+(Δy)^2)<=r_A+r_B

$\sqrt{{\left(3 - \left(- 2\right)\right)}^{2} + {\left(2 - 1\right)}^{2}} \le 6 + 3$

$\sqrt{25 + 1} \le 9$

$\sqrt{26} \le 9$

The last statement is true. Therefore the two circles overlap.