# Circle A has a center at (3 ,1 ) and a radius of 4 . Circle B has a center at (9 ,8 ) and a radius of 2 . Do the circles overlap? If not what is the smallest distance between them?

Apr 9, 2017

no overlap, min distance$\approx 3.22$

#### Explanation:

What we have to do here is $\textcolor{b l u e}{\text{compare}}$ the distance ( d ) between the centres of the circles to the $\textcolor{b l u e}{\text{sum of radii}}$

• "If sum of radii "> d" then circles overlap"

• "If sum of radii "< d" then no overlap"

To calculate d use the $\textcolor{b l u e}{\text{distance formula}}$

$\textcolor{red}{\overline{\underline{| \textcolor{w h i t e}{\frac{2}{2}} \textcolor{b l a c k}{d = \sqrt{{\left({x}_{2} - {x}_{1}\right)}^{2} + {\left({y}_{2} - {y}_{1}\right)}^{2}}} \textcolor{w h i t e}{\frac{2}{2}} |}}}$
where$\left({x}_{1} , {y}_{1}\right) , \left({x}_{2} , {y}_{2}\right) \text{ are 2 coordinate points}$

$\text{the 2 points here are " (3,1)" and } \left(9 , 8\right)$

$\text{let " (x_1,y_1)=(3,1)" and } \left({x}_{2} , {y}_{2}\right) = \left(9 , 8\right)$

$d = \sqrt{{\left(9 - 3\right)}^{2} + {\left(8 - 1\right)}^{2}} = \sqrt{36 + 49} = \sqrt{85} \approx 9.22$

$\text{sum of radii } = 4 + 2 = 6$

$\text{since sum of radii "< d" then no overlap}$

$\text{min. distance between circles "=d-" sum of radii}$

$\Rightarrow \text{smallest distance } = 9.22 - 6 = 3.22$
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