# Circle A has a center at (3 ,7 ) and a radius of 2 . Circle B has a center at (4 ,-2 ) and a radius of 6 . Do the circles overlap? If not, what is the smallest distance between them?

Dec 20, 2017

$\text{no overlap } , \approx 1.055$

#### Explanation:

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

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

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

$\text{to calculate d use the "color(blue)"distance formula}$

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

$\text{let "(x_1,y_1)=(3,7)" and } \left({x}_{2} , {y}_{2}\right) = \left(4 , - 2\right)$

$d = \sqrt{{\left(4 - 3\right)}^{2} + {\left(- 2 - 7\right)}^{2}} = \sqrt{82} \approx 9.055$

$\text{sum of radii } = 2 + 6 = 8$

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

$\text{smallest distance "=d-" sum of radii}$

$\textcolor{w h i t e}{\text{smallest distance}} = 9.055 - 8 = 1.055$
graph{((x-3)^2+(y-7)^2-4)((x-4)^2+(y+2)^2-36)=0 [-20, 20, -10, 10]}