# Circle A has a center at (12 ,9 ) and an area of 25 pi. Circle B has a center at (3 ,1 ) and an area of 67 pi. Do the circles overlap?

Aug 7, 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

Before doing this, we require to find the radii of both circles.

$\textcolor{\mathmr{and} a n \ge}{\text{ Reminder}}$ the area(A) of a circle is

$\textcolor{red}{| \overline{\underline{\textcolor{w h i t e}{\frac{a}{a}} \textcolor{b l a c k}{A = \pi {r}^{2}} \textcolor{w h i t e}{\frac{a}{a}} |}}}$

$\textcolor{b l u e}{\text{Circle A }} \pi {r}^{2} = 25 \pi \Rightarrow {r}^{2} = \frac{25 \cancel{\pi}}{\cancel{\pi}} \Rightarrow r = 5$

color(blue)"Circle B " pir^2=67pirArrr^2=67rArrr=sqrt67≈8.185

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 $\left({x}_{1} , {y}_{1}\right) \text{ and " (x_2,y_2)" are 2 coordinate points}$

Here the 2 points are (12 ,9) and (3 ,1) the centres of the circles.

let $\left({x}_{1} , {y}_{1}\right) = \left(12 , 9\right) \text{ and } \left({x}_{2} , {y}_{2}\right) = \left(3 , 1\right)$

d=sqrt((3-12)^2+(1-9)^2)=sqrt(81+64)=sqrt145≈12.042

sum of radii = radius of A + radius of B = 5 + 8.185 = 13.185

Since sum of radii > d , then circles overlap
graph{(y^2-18y+x^2-24x+200)(y^2-2y+x^2-6x-57)=0 [-41.1, 41.1, -20.55, 20.55]}