# Circle A has a center at (1 ,8 ) and an area of 15 pi. Circle B has a center at (5 ,3 ) and an area of 25 pi. Do the circles overlap?

Mar 27, 2016

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 $\left({x}_{1} , {y}_{1}\right) = \left(1 , 8\right) \text{ and } \left({x}_{2} , {y}_{2}\right) = \left(5 , 3\right)$

→ d = sqrt((5-1)^2+(3-8)^2) = sqrt(16+25) = sqrt41 ≈ 6.403

Now , require to find the radii of the circles.Given the area , we can calculate r , using
area of circle $= \pi {r}^{2}$

circle A : $\pi {r}^{2} = 15 \pi \Rightarrow {r}^{2} = \frac{15 \pi}{\pi} = 15 \Rightarrow r = \sqrt{15}$

circle B : $\pi {r}^{2} = 25 \pi \Rightarrow {r}^{2} = 25 \Rightarrow r = \sqrt{25} = 5$

radius of A + radius of B =  sqrt15 + 5 ≈ 8.873

sum of radii > distance between centres , hence they overlap.