Circle A has a radius of #2 # and a center of #(5 ,7 )#. Circle B has a radius of #4 # and a center of #(7 ,2 )#. If circle B is translated by #<2 ,-5 >#, does it overlap circle A? If not, what is the minimum distance between points on both circles?
1 Answer
no overlap, ≈ 4.77
Explanation:
To determine this , we require to compare the distance (d) between the centres of the circles with the sum of their radii.
• If sum of radii > d , then circles will overlap.
• If sum of radii < d , then no overlap will occur.
A translation does not change the shape of a figure , only it's position.
Under a translation of
# ((2),(-5))# centre of B (7 , 2) → (7+2 ,2-5 ) → (9 , -3)
To calculate the distance between the 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 coordinate points " # let
# (x_1,y_1)=(5 , 7)" and " (x_2,y_2)=(9 , -3) #
#rArr d = sqrt((9-5)^2 + (-3-7)^2) =sqrt(16+100) ≈ 10.77# now, radius of A + radius of B = 2 + 4 = 6
Since sum of radii < d , no overlap
and distance between circles = 10.77 - 6 = 4.77
