# Circle A has a center at (1 ,5 ) and a radius of 3 . Circle B has a center at (2 ,-3 ) and a radius of 5 . Do the circles overlap? If not, what is the smallest distance between them?

Feb 23, 2016

They do not overlap.

#### Explanation:

To overlap the distance between centres has to be less than the sum of their radii.

Let the sum of radii be $r$
Let the distance between radii be $d$

$\text{Centre"_"A} \to \left({x}_{1} , {y}_{1}\right) \to \left(1 , 5\right)$
$\text{Centre"_"B} \to \left({x}_{2} , {y}_{2}\right) \to \left(2 , - 3\right)$

Sum of radii $\to 3 + 5 = 8 = r$

$\textcolor{b l u e}{\text{Distance between centres}}$

Using Pythagoras

${d}^{2} = {\left({x}_{2} - {x}_{1}\right)}^{2} + {\left({y}_{2} - {y}_{1}\right)}^{2}$

$d = \pm \sqrt{{\left(2 - 1\right)}^{2} + {\left(- 3 - 5\right)}^{2}}$

$d = \pm \sqrt{{1}^{2} + {\left(- 8\right)}^{2}} \text{ "=" } \pm 8.062$ to 3 decimal places

As we are comparing sizes the $\pm$ does not have any significance.

As $d > s$ the do not overlap