Circle A has a radius of #2 # and a center of #(7 ,6 )#. Circle B has a radius of #3 # and a center of #(5 ,3 )#. If circle B is translated by ##, does it overlap circle A? If not, what is the minimum distance between points on both circles?

Question

Circle A has a radius of #2 # and a center of #(7 ,6 )#.  Circle B has a radius of #3 # and a center of #(5 ,3 )#.  If circle B is translated by #<-1 ,2 >#, does it overlap circle A?  If not, what is the minimum distance between points on both circles?

Sophia Alfred · Accepted Answer

circles overlap

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]}

Violet Aldrich · Answer

undefined To determine if circle B overlaps circle A after being translated, calculate the distance between the centers of the circles after the translation. If this distance is less than the sum of the radii of both circles, then the circles overlap. Otherwise, they do not overlap. If the circles do not overlap, the minimum distance between points on both circles is equal to the difference between the distance between the centers and the sum of their radii.

Circle A has a radius of #2 # and a center of #(7 ,6 )#. Circle B has a radius of #3 # and a center of #(5 ,3 )#. If circle B is translated by #&lt;-1 ,2 &gt;#, does it overlap circle A? If not, what is the minimum distance between points on both circles?

Circle A has a radius of #2 # and a center of #(7 ,6 )#. Circle B has a radius of #3 # and a center of #(5 ,3 )#. If circle B is translated by #<-1 ,2 >#, does it overlap circle A? If not, what is the minimum distance between points on both circles?