Circle A has a radius of #3 # and a center of #(5 ,9 )#. Circle B has a radius of #4 # and a center of #(1 ,2 )#. 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 #3 # and a center of #(5 ,9 )#.  Circle B has a radius of #4 # and a center of #(1  ,2 )#.  If circle B is translated by #<3 ,2 >#, does it overlap circle A?  If not, what is the minimum distance between points on both circles?

Claire Akin · Accepted Answer

#R_A + R_B# #color(green)((7))# #># #vec(O_AO_B)# #color(blue)((4.123))#, the two circles A & B overlap.

Circle A #O_A (5,9), R_A = 3#, Circle B #O_B (1,2), R_B = 4#

#O_B# #translated# #by# #(3,2)#

New #O_B = ((1+3),(2+3)) => ((4),(5))#

Distance #vec(O_AO_B) = sqrt((4-5)^2 + (5-9)^2) = sqrt17 ~~ 4.123#

Sum of radii #R_A + R_B = 3 + 4 = 7#

Since #R_A + R_B# #color(green)((7))# #># #vec(O_AO_B)# #color(blue)((4.123))#, the two circles A & B overlap.

Violet Aldrich · Answer

undefined To determine if Circle B overlaps Circle A after the translation, calculate the distance between the centers of the circles after the translation. Then compare this distance with the sum of the radii of both circles. If the distance between the centers is greater than the sum of the radii, the circles do not overlap.

After the translation by <3, 2>, the new center of Circle B becomes (1 + 3, 2 + 2) = (4, 4).

Now, calculate the distance between the centers of the circles:

Distance = √((x2 - x1)^2 + (y2 - y1)^2)

Distance = √((5 - 4)^2 + (9 - 4)^2)
Distance = √(1^2 + 5^2)
Distance = √26 ≈ 5.099

The sum of the radii of both circles is 3 + 4 = 7.

Since the distance between the centers (5.099) is greater than the sum of the radii (7), the circles do not overlap.

To find the minimum distance between points on both circles, subtract the distance between the centers from the sum of the radii:

Minimum distance = 7 - 5.099 ≈ 1.901 units.

Circle A has a radius of #3 # and a center of #(5 ,9 )#. Circle B has a radius of #4 # and a center of #(1 ,2 )#. If circle B is translated by #&lt;3 ,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 #3 # and a center of #(5 ,9 )#. Circle B has a radius of #4 # and a center of #(1 ,2 )#. If circle B is translated by #<3 ,2 >#, does it overlap circle A? If not, what is the minimum distance between points on both circles?