# Circle A has a center at (1 ,2 ) and an area of 100 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?

Jul 29, 2016

circles overlap

#### Explanation:

What we have to do here is compare the distance ( d) between the centres of the circles to the sum of the radii.

• If sum of radii > d , then circles overlap

• If sum of radii < d , then no overlap

Before this can be done , we require to find the radii of both circles.

$\textcolor{\mathmr{and} a n \ge}{\text{Reminder}}$ the area (A) of a circle is.

$\textcolor{red}{| \overline{\underline{\textcolor{w h i t e}{\frac{a}{a}} \textcolor{b l a c k}{A = \pi {r}^{2}} \textcolor{w h i t e}{\frac{a}{a}} |}}}$

$\textcolor{b l u e}{\text{Circle A }} \pi {r}^{2} = 100 \pi \Rightarrow {r}^{2} = \frac{100 \cancel{\pi}}{\cancel{\pi}} \Rightarrow r = 10$

$\textcolor{b l u e}{\text{Circle B }} \pi {r}^{2} = 36 \pi \Rightarrow {r}^{2} = \frac{36 \cancel{\pi}}{\cancel{\pi}} \Rightarrow r = 6$

To calculate d, use the $\textcolor{b l u e}{\text{distance formula}}$

$\textcolor{red}{| \overline{\underline{\textcolor{w h i t e}{\frac{a}{a}} \textcolor{b l a c k}{d = \sqrt{{\left({x}_{2} - {x}_{1}\right)}^{2} + {\left({y}_{2} - {y}_{1}\right)}^{2}}} \textcolor{w h i t e}{\frac{a}{a}} |}}}$
where $\left({x}_{1} , {y}_{1}\right) \text{ and " (x_2,y_2)" are 2 coordinate points}$

here the 2 points are (1 ,2) and (7 ,9) the centres of the circles.

let $\left({x}_{1} , {y}_{1}\right) = \left(1 , 2\right) \text{ and } \left({x}_{2} , {y}_{2}\right) = \left(7 , 9\right)$

d=sqrt((7-1)^2+(9-2)^2)=sqrt(36+49)=sqrt85≈9.22

sum of radii = radius of A + radius of B = 10 + 6 = 16

Since sum of radii > d , then circles overlap
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