# Circle A has a center at (11 ,5 ) and an area of 100 pi. Circle B has a center at (4 ,9 ) and an area of 36 pi. Do the circles overlap? If not, what is the shortest distance between them?

Mar 21, 2016

circles overlap

#### Explanation:

First step is to find 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 $\left({x}_{1} , {y}_{1}\right) = \left(11 , 5\right) \text{ and } \left({x}_{2} , {y}_{2}\right) = \left(4 , 9\right)$

substitute these values into the formula to find d

 d = sqrt((4-11)^2 + (9-5)^2) = sqrt(49+16) = sqrt65 ≈ 8.06

Now, require to find the radii of circles A and B, given that the areas are known.

Circle A : $\pi {r}^{2} = 100 \pi \Rightarrow {r}^{2} = \frac{100 \pi}{\pi} = 100 \Rightarrow r = 10$

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

radius of circle A + radius of circle B = 10 + 6 = 16

since sum of radii > distance between centres → overlap