# Circle A has a radius of 4  and a center of (7 ,3 ). Circle B has a radius of 2  and a center of (2 ,2 ). If circle B is translated by <1 ,3 >, does it overlap circle A? If not, what is the minimum distance between points on both circles?

Feb 9, 2018

circles overlap

#### Explanation:

${O}_{A} \left(\begin{matrix}7 \\ 3\end{matrix}\right) , {R}_{A} = 4 , {O}_{B} \left(\begin{matrix}2 \\ 2\end{matrix}\right) , {R}_{B} = 2$

${O}_{B} t r a n s l a t e d b y \left(\begin{matrix}1 \\ 3\end{matrix}\right)$

New coordinates of ${O}_{B} \to \left(\begin{matrix}2 + 1 \\ 2 + 3\end{matrix}\right) \implies \left(\begin{matrix}3 \\ 5\end{matrix}\right)$

$\vec{{O}_{A} {O}_{B}} = \sqrt{{\left(7 - 3\right)}^{2} + {\left(3 - 5\right)}^{2}} = \sqrt{20} \approx 4.47$

Sum of Radii ${R}_{A} + {R}_{B} = 4 + 2 = 6$

${R}_{A} + {R}_{B} > \vec{{O}_{A} {O}_{B}}$ and both the circles overlap