Question

Circle A has a radius of `2` and a center at `(5 ,6 )`. Circle B has a radius of `5` and a center at `(2 ,4 )`. If circle B is translated by `<-2 ,1 >`, does it overlap circle A? If not, what is the minimum distance between points on both circles?

Answer 1

circles overlap

Explanation:

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

• If sum of radii > d , then circles overlap

• If sum of radii < d , then no overlap

However we require to find the new centre of B under the translation.
A translation does not change the shape of a figure only it's position.

Under a translation `((-2),(1))`

centre B(2 ,4) → (-2+2 ,4+1) → (0 ,5)

To calculate the distance (d) between centres 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 points"`

let `(x_1,y_1)=(5,6)" and " (x_2,y_2)=(0,5)`

`rArr d=sqrt((0-5)^2+(5-6)^2)=sqrt26 ≈ 5.099`

radius of A + radius of B = 2 + 5 = 7

Since sum of radii > d , then circles overlap
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