# Circle A has a center at (2 ,5 ) and a radius of 3 . Circle B has a center at (7 ,2 ) and a radius of 3 . Do the circles overlap? If not, what is the smallest distance between them?

Aug 31, 2016

circles overlap.

#### Explanation:

What we have to do here is compare the distance (d) between the centres of the circles to 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 $\left({x}_{1} , {y}_{1}\right) \text{ and " (x_2,y_2)" are 2 coordinate points}$

The 2 points here are (2 ,5) and (7 ,2) the centres of the circles.

let $\left({x}_{1} , {y}_{1}\right) = \left(2 , 5\right) \text{ and } \left({x}_{2} , {y}_{2}\right) = \left(7 , 2\right)$

d=sqrt((7-2)^2+(2-5)^2)=sqrt(25+9)=sqrt34≈5.831

sum of radii = radius of A + radius of B = 3 + 3 = 6

Since sum of radii > d , then circles overlap
graph{(y^2-10y+x^2-4x+20)(y^2-4y+x^2-14x+44)=0 [-24.98, 24.97, -12.5, 12.46]}