# Circle A has a center at (2 ,4 ) and a radius of 5 . Circle B has a center at (9 ,3 ) and a radius of 1 . Do the circles overlap? If not what is the smallest distance between them?

Dec 8, 2017

$\text{no overlap } , \approx 1.07$

#### 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 $\textcolor{b l u e}{\text{sum of the radii}}$

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

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

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

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

$\text{let "(x_1,y_1)=(2,4)" and } \left({x}_{2} , {y}_{2}\right) = \left(9 , 3\right)$

$d = \sqrt{{\left(9 - 2\right)}^{2} + {\left(3 - 4\right)}^{2}} = \sqrt{49 + 1} = \sqrt{50} \approx 7.07$

$\text{sum of radii } = 5 + 1 = 6$

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

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

$\textcolor{w h i t e}{\times \times \times \times \times \times} = 7.07 - 6 = 1.07$
graph{((x-2)^2+(y-4)^2-25)((x-9)^2+(y-3)^2-1)=0 [-20, 20, -10, 10]}