Question

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?

Answer 1

The distance `d(A,B)` and the radius of each circle `r_A` and `r_B` must satisfy the condition:

`d(A,B)<=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(A,B)` 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(A,B)<=r_A+r_B`

The Euclidean distance `d(A,B)` can be calculated:

`d(A,B)=sqrt((Δx)^2+(Δy)^2)`

Therefore:

`d(A,B)<=r_A+r_B`

`sqrt((Δx)^2+(Δy)^2)<=r_A+r_B`

`sqrt((3-(-2))^2+(2-1)^2)<=6+3`

`sqrt(25+1)<=9`

`sqrt(26)<=9`

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