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

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Feb 9, 2018

Circles A and B overlap.

#### Explanation:

Circle A - $C e n t e r {O}_{A} \left(5 , 3\right) , {R}_{A} = 4$

Circle A - $C e n t e r {O}_{A} \left(1 , 4\right) , {R}_{A} = 3$

Circle B translated by (2,4)

New center of B ${O}_{B} \left(\begin{matrix}1 + 2 \\ 4 + 4\end{matrix}\right) \implies \left(\begin{matrix}3 \\ 8\end{matrix}\right)$

$\vec{{O}_{A} {O}_{B}} = \sqrt{{\left(5 - 3\right)}^{2} + {\left(3 - 8\right)}^{2}} = \sqrt{29} \approx 5.39$

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

Since ${R}_{A} + {R}_{B} > \vec{{O}_{A} {O}_{B}}$, circles A and B overlap.

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