Circle A has a center at #(5 ,2 )# and an area of #18 pi#. Circle B has a center at #(3 ,6 )# and an area of #27 pi#. Do the circles overlap?

Question

Circle A has a center at #(5   ,2  )# and an area of #18 pi#.  Circle B has a center at #(3  ,6 )# and an area of #27 pi#.  Do the circles overlap?

Savannah Aguilar · Accepted Answer

The circles do overlap (by #4.87# units)

Since Circle A has an Area #=18pi#, Circle A has a Radius of #r_A = sqrt(18) = 3sqrt(2)~~4.24# #color(white)("XXX")#(this follows from formua #"Area" =pir^2#) Since Circle B has an Area #=27pi# Circle B has a Radius of #r_B=sqrt(27)=3sqrt(3)~~5.20# The distance between the center of A at #(5,2)# and the center of B at #(3,6)# is #color(white)("XXX")d=sqrt((5-3)^2+(2-6)^2)=2sqrt(5)~~4.47# Together the radii of circles A and B cover #4.24+5.20 = 9.34# of the line segment joining the centers of A and B. Since this is greater than the actual length of the line segment joining A and B, the radii must overlap by #9.34-4.47=4.87# units. graph{((x-5)^2+(y-2)^2-18)((x-3)^2+(y-6)^2-27)=0 [-14.11, 17.94, -3.74, 12.28]}

Sadie Adams · Answer

undefined To determine if the circles overlap, we need to find the distance between their centers and compare it to the sum of their radii. If the distance between the centers is less than the sum of their radii, the circles overlap.

The distance between two points (x1, y1) and (x2, y2) is given by the formula:

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

For Circle A, the center is (5, 2), and for Circle B, the center is (3, 6).

Distance between the centers:
√((3 - 5)^2 + (6 - 2)^2) = √((-2)^2 + (4)^2) = √(4 + 16) = √20 ≈ 4.47

The radius of Circle A is √(Area of Circle A / π) = √(18π / π) = √18 ≈ 4.24
The radius of Circle B is √(Area of Circle B / π) = √(27π / π) = √27 ≈ 5.20

Sum of the radii: 4.24 + 5.20 ≈ 9.44

Since the distance between the centers (4.47) is less than the sum of the radii (9.44), the circles overlap.