# 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?

Oct 17, 2017

They do not overlap, minimum distance is $\sqrt{65} - 7 \approx 1.06$.

#### 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.

$\left(4 , 5\right) + \left(- 3 , 4\right)$
$\left(4 - 3 , 5 + 4\right)$
$\left(1 , 9\right)$

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.

$d = \sqrt{{\left(5 - 1\right)}^{2} + {\left(2 - 9\right)}^{2}}$

$d = \sqrt{{\left(4\right)}^{2} + {\left(- 7\right)}^{2}}$

$d = \sqrt{16 + 49}$

$d = \sqrt{65} \approx 8.06$

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.

$\sqrt{65} - 7 \approx 1.06$