# Circle A has a center at (5 ,2 ) and a radius of 4 . Circle B has a center at (9 ,8 ) and a radius of 7 . Do the circles overlap? If not what is the smallest distance between them?

Jun 16, 2016

circles overlap

#### Explanation:

What we have to do here is compare the distance (d ) between the centres with the sum of the radii.

• If sum of radii > d , then circles overlap

• If sum of radii < d , then no overlap

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

$\textcolor{red}{| \overline{\underline{\textcolor{w h i t e}{\frac{a}{a}} \textcolor{b l a c k}{d = \sqrt{{\left({x}_{2} - {x}_{1}\right)}^{2} + {\left({y}_{2} - {y}_{1}\right)}^{2}}} \textcolor{w h i t e}{\frac{a}{a}} |}}}$
where $\left({x}_{1} , {y}_{1}\right) \text{ and " (x_2,y_2)" are 2 points}$

let $\left({x}_{1} , {y}_{1}\right) = \left(5 , 2\right) \text{ and } \left({x}_{2} , {y}_{2}\right) = \left(9 , 8\right)$

d=sqrt((9-5)^2+(8-2)^2)=sqrt(16+36)≈7.211

radius of A + radius of B = 4 + 7 = 11

Since sum of radii > d , then circles overlap
graph{(y^2-4y+x^2-10x+13)(y^2-16y+x^2-18x+96)=0 [-40, 40, -20, 20]}