Circle A has a radius of 2  and a center at (5 ,6 ). Circle B has a radius of 5  and a center at (3 ,4 ). If circle B is translated by <-2 ,1 >, does it overlap circle A? If not, what is the minimum distance between points on both circles?

Sep 28, 2016

The circles overlap.

Explanation:

If circle $B$ with center $\left(3 , 4\right)$ is translated by $< - 2 , 1 >$ its new center will be at $\left(3 - 2 , 4 + 1\right) = \left(1 , 5\right)$
The distance between the center of $A$ at $\left(5 , 6\right)$ and the new center of $B$ is given by the Pythagorean Theorem as:
$\textcolor{w h i t e}{\text{XXX}} d = \sqrt{{\left(5 - 1\right)}^{2} + {\left(6 - 5\right)}^{2}} = \sqrt{17} \approx 4.123$

We are told that circle $A$ has a radius of $2$ and circle $B$ has a radius of $2$ and circle $B$ has a radius of $5$.

Considering the line segment joining the centers of the two circles,
we can see that $A$ covers $2$ units of that line segment and $B$ covers $5$ units (actually more than the length of the line segment.

Since there is only $4.123$ units between the two centers, the circles must overlap. 