Circle A has a radius of #4 # and a center at #(7 ,2 )#. Circle B has a radius of #3 # and a center at #(6 ,5 )#. 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 #4 # and a center at #(7 ,2 )#.  Circle B has a radius of #3 # and a center at #(6 ,5 )#.  If circle B is translated by #<-3 ,4 >#, does it overlap circle A?  If not, what is the minimum distance between points on both circles?

Autumn Armstrong · Accepted Answer

no overlap , ≈ 1.062

What we have to do here is #color(blue)"compare"# the distance ( d) between the centres of the circles to the #color(blue)"sum of the radii".# • If sum of radii > d , then circles overlap • If sum of radii < d , then no overlap However, the first step here is to calculate the new centre of B under the given translation, which does not change the shape of the circle only it's position. Under a translation #((-3),(4))# #B(6,5)to(6-3,5+4)to(3,9)" new centre of B"# 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 (7 ,2) and (3 ,9) the centres of the circles. let #(x_1,y_1)=(7,2)" and " (x_2,y_2)=(3,9)# #d=sqrt((3-7)^2+(9-2)^2)=sqrt(16+49)=sqrt65≈8.062# sum of radii = radius of A + radius of B = 4 + 3 = 7 Since sum of radii < d , then no overlap of circles min. distance = d - sum of radii = 8.062 - 7 = 1.062 graph{(y^2-4y+x^2-14x+37)(y^2-18y+x^2-6x+81)=0 [-40, 40, -20, 20]}

Noah Atkinson · Answer

undefined To determine if circle B overlaps circle A after being translated, we need to find the distance between the centers of the two circles after the translation and compare it to the sum of their radii. If the distance between the centers is less than the sum of the radii, the circles overlap; otherwise, they do not.

First, we find the new center of circle B after translation:
New center of B = (6 - 3, 5 + 4) = (3, 9)

Next, we calculate the distance between the centers of circles A and B:
Distance = √((x2 - x1)^2 + (y2 - y1)^2)
Distance = √((7 - 3)^2 + (2 - 9)^2)
Distance ≈ √((4)^2 + (-7)^2)
Distance ≈ √(16 + 49)
Distance ≈ √65 ≈ 8.06

Now, we compare the distance between the centers to the sum of the radii:
Sum of radii = 4 + 3 = 7

Since the distance between the centers (8.06) is greater than the sum of the radii (7), the circles do not overlap.

To find the minimum distance between points on both circles, we subtract the distance between the centers from the sum of the radii:
Minimum distance = Distance - Sum of radii
Minimum distance ≈ 8.06 - 7 ≈ 1.06

Therefore, the minimum distance between points on both circles is approximately 1.06 units.

Circle A has a radius of #4 # and a center at #(7 ,2 )#. Circle B has a radius of #3 # and a center at #(6 ,5 )#. If circle B is translated by #&lt;-3 ,4 &gt;#, does it overlap circle A? If not, what is the minimum distance between points on both circles?

Circle A has a radius of #4 # and a center at #(7 ,2 )#. Circle B has a radius of #3 # and a center at #(6 ,5 )#. If circle B is translated by #<-3 ,4 >#, does it overlap circle A? If not, what is the minimum distance between points on both circles?