# Circle A has a center at (1 ,3 ) and an area of 16 pi. Circle B has a center at (2 ,7 ) and an area of 75 pi. Do the circles overlap?

Mar 24, 2016

overlap

#### Explanation:

The 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 $\left({x}_{1} , {y}_{1}\right) \mathmr{and} \left({x}_{2} , {y}_{2}\right) \text{ are 2 coordinate points }$

let $\left({x}_{1} , {y}_{1}\right) = \left(1 , 3\right) \text{ and } \left({x}_{2} , {y}_{2}\right) = \left(2 , 7\right)$

rArr d = sqrt((2-1)^2 + (7-3)^2) = sqrt(1+16) = sqrt17 ≈ 4.123

Now require to find the radii of the circles.

Using : area of circle = $\pi {r}^{2} \text{ where r is the radius }$

Circle A : $\pi {r}^{2} = 16 \pi \Rightarrow {r}^{2} = 16 \Rightarrow r = 4$

Circle B :  pir^2 = 75pi rArr r^2 = 75rArr r = sqrt75 ≈ 8.66

radius of A + radius of B = 4 + 8.66 = 10.66

Since sum of radii > distance between centres → overlap