# Circle A has a center at (5 ,4 ) and an area of 15 pi. Circle B has a center at (2 ,1 ) and an area of 90 pi. Do the circles overlap?

Sep 29, 2017

$\text{one circle inside other}$

#### Explanation:

What we have to do here is $\textcolor{b l u e}{\text{compare }}$ the distance (d) between the centres of the circles to the $\textcolor{b l u e}{\text{sum/difference}}$ of the radii.

• " if sum of radii ">d" then circles overlap"

• " if sum of radii"< d" then no overlap"

• " if difference of radii">d" then circle inside other"

• " area of circle "=pir^2larr" r is the radius"

$\textcolor{b l u e}{\text{Circle A}}$

$\Rightarrow \pi {r}^{2} = 15 \pi \Rightarrow r = \sqrt{15} \approx 3.873$

$\textcolor{b l u e}{\text{Circle B}}$

$\Rightarrow \pi {r}^{2} = 90 \pi \Rightarrow r = \sqrt{90} \approx 9.486$

$\text{to calculate d use the "color(blue)"gradient formula}$

$\textcolor{red}{\overline{\underline{| \textcolor{w h i t e}{\frac{2}{2}} \textcolor{b l a c k}{d = \sqrt{{\left({x}_{2} - {x}_{1}\right)}^{2} + {\left({y}_{2} - {y}_{1}\right)}^{2}}} \textcolor{w h i t e}{\frac{2}{2}} |}}}$

$\text{let "(x_1,y_1)=(2,1)" and } \left({x}_{2} , {y}_{2}\right) = \left(5 , 4\right)$

$d = \sqrt{{\left(5 - 2\right)}^{2} + {\left(4 - 1\right)}^{2}} = \sqrt{18} \approx 4.243$

$\text{sum of radii } = 3.873 + 9.486 = 13.359$

$\text{difference of radii } = 9.486 - 3.873 = 5.613$

$\text{since difference of radii">d" then circle inside other}$
graph{((x-5)^2+(y-4)^2-15)((x-2)^2+(y-1)^2-90)=0 [-40, 40, -20, 20]}