# Circle A has a radius of 3  and a center of (5 ,9 ). Circle B has a radius of 4  and a center of (1 ,2 ). If circle B is translated by <3 ,2 >, does it overlap circle A? If not, what is the minimum distance between points on both circles?

Feb 9, 2018

${R}_{A} + {R}_{B}$ $\textcolor{g r e e n}{\left(7\right)}$ $>$ $\vec{{O}_{A} {O}_{B}}$ $\textcolor{b l u e}{\left(4.123\right)}$, the two circles A & B overlap.

#### Explanation:

Circle A ${O}_{A} \left(5 , 9\right) , {R}_{A} = 3$, Circle B ${O}_{B} \left(1 , 2\right) , {R}_{B} = 4$

${O}_{B}$ $t r a n s l a t e d$ $b y$ $\left(3 , 2\right)$

New ${O}_{B} = \left(\begin{matrix}1 + 3 \\ 2 + 3\end{matrix}\right) \implies \left(\begin{matrix}4 \\ 5\end{matrix}\right)$

Distance $\vec{{O}_{A} {O}_{B}} = \sqrt{{\left(4 - 5\right)}^{2} + {\left(5 - 9\right)}^{2}} = \sqrt{17} \approx 4.123$

Sum of radii ${R}_{A} + {R}_{B} = 3 + 4 = 7$

Since ${R}_{A} + {R}_{B}$ $\textcolor{g r e e n}{\left(7\right)}$ $>$ $\vec{{O}_{A} {O}_{B}}$ $\textcolor{b l u e}{\left(4.123\right)}$, the two circles A & B overlap.