# Circle A has a radius of 2  and a center of (2 ,7 ). Circle B has a radius of 1  and a center of (3 ,1 ). 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?

Dec 6, 2016

$\text{1- no overlap}$
$\text{2-minimum distance is } 0.12$

#### Explanation:

$\text{1-draw circles given}$

$\text{2-Translate circle B by (1,3)}$

$\text{the new position of circle B is shown in figure below}$

$\text{now find distance between "O_1" and } {O}_{T}$

${O}_{1} {O}_{T} = \sqrt{{\left(2 - 1\right)}^{2} + {\left(7 - 3\right)}^{2}}$

${O}_{1} {O}_{T} = \sqrt{{1}^{2} + {4}^{2}} = \sqrt{1 + 16} = \sqrt{17} = 4.12$

$\text{radius } {r}_{1} = 2$

$\text{radius } {r}_{2} = 1$

${r}_{1} + {r}_{2} = 2 + 1 = 3$

${O}_{1} {O}_{T} > {r}_{1} + {r}_{2} \text{ ; no overlap}$

$\text{minimum distance is AB line segment(colored blue)}$

$A B = {O}_{1} {O}_{T} - {r}_{1} + {r}_{2}$

$A B = 4.12 - 3$

$A B = 0.12$