Circle A has a radius of 2 2 and a center of (7 ,6 )(7,6). Circle B has a radius of 3 3 and a center of (5 ,3 )(5,3). If circle B is translated by <-1 ,2 ><−1,2>, does it overlap circle A? If not, what is the minimum distance between points on both circles?
1 Answer
circles overlap
Explanation:
What we have to do here is 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
The first step is to find the new centre of B under the given translation. Under a translation the shape of the figure does not change only it's position.
Under a translation
((-1),(2)) centre of B (5 ,3) → (5-1 ,3+2) → (4 ,5)
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 points" The 2 points here are (7 ,6) and (4 ,5)
d=sqrt((4-7)^2+(5-6)^2)=sqrt10≈3.162 radius of A + radius of B = 2 + 3 = 5
Since sum of radii > d , then circles overlap.
graph{(y^2-12y+x^2-14x+81)(y^2-10y+x^2-8x+32)=0 [-20, 20, -10, 10]}
