Circle A has a center at #(1 ,3 )# and an area of #16 pi#. Circle B has a center at #(2 ,7 )# and an area of #75 pi#. Do the circles overlap?

Question

Circle A has a center at #(1   ,3  )# and an area of #16 pi#.  Circle B has a center at #(2  ,7   )# and an area of #75 pi#.  Do the circles overlap?

Mason Adams · Accepted Answer

overlap

The first step is to calculate the distance between the centres using the #color(blue)" distance formula "# # d = sqrt((x_2 - x_1)^2 + (y_2 - y_1)^2) # where # (x_1,y_1) and (x_2,y_2)" are 2 coordinate points "# let # (x_1,y_1)=(1,3)" and " (x_2,y_2)=(2,7)# #rArr d = sqrt((2-1)^2 + (7-3)^2) = sqrt(1+16) = sqrt17 ≈ 4.123# Now require to find the radii of the circles. Using : area of circle = #pir^2" where r is the radius " # Circle A : #pir^2 = 16pi rArr r^2 = 16 rArr r = 4 # Circle B : # pir^2 = 75pi rArr r^2 = 75rArr r = sqrt75 ≈ 8.66# radius of A + radius of B = 4 + 8.66 = 10.66 Since sum of radii > distance between centres → overlap

Evelyn Arboleda · Answer

undefined To determine if the circles overlap, calculate 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; otherwise, they do not overlap.

The distance between the centers of circles A and B can be found using the distance formula:

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

For circle A (center at (1, 3)) and circle B (center at (2, 7)):

Distance = √((2 - 1)^2 + (7 - 3)^2)
         = √(1^2 + 4^2)
         = √(1 + 16)
         = √17

The radii of the circles are the square roots of their respective areas divided by pi:

Radius of circle A = √(16π/pi) = √16 = 4
Radius of circle B = √(75π/pi) = √75 ≈ 8.66

The sum of the radii of circles A and B is 4 + 8.66 = 12.66.

Since the distance between the centers (√17) is less than the sum of their radii (12.66), the circles overlap.