# Circle A has a center at (1 ,4 ) and an area of 28 pi. Circle B has a center at (7 ,9 ) and an area of 36 pi. Do the circles overlap? If not, what is the shortest distance between them?

Mar 12, 2016

circles overlap

#### Explanation:

First step is to calculate the distance between the centres using the$\textcolor{b l u e}{\text{ distance formula }}$
$d = \sqrt{{\left({x}_{2} - {x}_{1}\right)}^{2} + {\left({y}_{2} - {y}_{1}\right)}^{2}}$

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

let $\left({x}_{1} , {y}_{1}\right) = \left(1 , 4\right) \text{ and } \left({x}_{2} , {y}_{2}\right) = \left(7 , 9\right)$
hence d = sqrt((7-1)^2 +(9-4)^2 )= sqrt(6^2+5^2)=sqrt61≈ 7.81

Now require to find radii of circles,using A = $\pi {r}^{2}$

circle A : $\pi {r}^{2} = 28 \pi \Rightarrow {r}^{2} = 28 \Rightarrow r = \sqrt{28}$

circle B : $\pi {r}^{2} = 36 \pi \Rightarrow {r}^{2} = 36 \Rightarrow r = \sqrt{36} = 6$

radius of A + radius of B = sqrt28 + 6 ≈ 11.292

since the sum of the radii > distance between centres , the circles will overlap