Question

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?

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 this can be done , 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=100pirArrr^2=(100cancel(pi))/cancel(pi)rArrr=10`

`color(blue)"Circle B " pir^2=36pirArrr^2=(36cancel(pi))/cancel(pi)rArrr=6`

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

let `(x_1,y_1)=(1,2)" and " (x_2,y_2)=(7,9)`

`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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