Question

Circle A has a center at `(12 ,9 )` and an area of `25 pi`. Circle B has a center at `(3 ,1 )` and an area of `67 pi`. Do the circles overlap?

Answer 1

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 doing this, we require to find the radii of both circles.

`color(orange)" Reminder"` the area(A) of a circle is

`color(red)(|bar(ul(color(white)(a/a)color(black)(A=pir^2)color(white)(a/a)|)))`

`color(blue)"Circle A " pir^2=25pirArrr^2=(25cancel(pi))/cancel(pi)rArrr=5`

`color(blue)"Circle B " pir^2=67pirArrr^2=67rArrr=sqrt67≈8.185`

To calculate d, use the `color(blue)"distance formula"`

`color(red)(|bar(ul(color(white)(a/a)color(black)(d=sqrt((x_2-x_1)^2+(y_2-y_1)^2))color(white)(a/a)|)))`
where `(x_1,y_1)" and " (x_2,y_2)" are 2 coordinate points"`

Here the 2 points are (12 ,9) and (3 ,1) the centres of the circles.

let `(x_1,y_1)=(12,9)" and " (x_2,y_2)=(3,1)`

`d=sqrt((3-12)^2+(1-9)^2)=sqrt(81+64)=sqrt145≈12.042`

sum of radii = radius of A + radius of B = 5 + 8.185 = 13.185

Since sum of radii > d , then circles overlap
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