Question

Circle A has a radius of `5` and a center of `(5 ,2 )`. Circle B has a radius of `2` and a center of `(4 ,5 )`. If circle B is translated by `<-3 ,4 >`, does it overlap circle A? If not, what is the minimum distance between points on both circles?

Answer 1

They do not overlap, minimum distance is `sqrt(65) - 7 ~~ 1.06`.

Explanation:

First let's move the center of circle B.
Translation simply moves the circle's center, so we can just add the translation amount to the x and y values of the center.

`(4,5) + (-3,4)`
`(4-3,5+4)`
`(1,9)`

So now we just need to figure out if A and B overlap.
B has a radius of 2, and A has a radius 5.
`5+2 = 7`
So, they will overlap if A's center and B's center are less than 7 away.

Distance from A's center to B's center can be calculated using distance formula.

`d = sqrt((5-1)^2 + (2-9)^2)`

`d = sqrt((4)^2 + (-7)^2)`

`d = sqrt(16 + 49)`

`d = sqrt(65) ~~8.06`

The distance is greater than `7`, so they don't intersect.
If you do not have a calculator to calculate the square root, we know that `sqrt(49) = 7`, so `sqrt(65)` must be greater than `7`.

The minimum distance between the circles can be calculated by taking the difference of the distance and 7.
This is because a line drawn from one center to the other will be the length of both radii plus the minimum distance between the circles.

`sqrt(65) - 7 ~~ 1.06`