Circle A has a center at #(1 ,8 )# and an area of #15 pi#. Circle B has a center at #(5 ,3 )# and an area of #24 pi#. Do the circles overlap?

Question

Circle A has a center at #(1   ,8  )# and an area of #15 pi#.  Circle B has a center at #(5  ,3   )# and an area of #24 pi#.  Do the circles overlap?

James Atkins · Accepted Answer

The distance between the centers is less than the sum of their radii, therefore, the circles overlap.

The area of a circle is #A = pir²#. This allows us to observe that the radius of circle A is: #r_a= sqrt15# And the radius of circle B is: #r_b = sqrt24#. The distance, d, between the centers is: #d = sqrt((1 - 5)² + (8 - 3)²)# #d = sqrt((-4)² + (5)²)# #d = sqrt(16 + 25)# #d = sqrt41# #d < (r_a + r_b)# by 2.369, therefore, the circles overlap

Aurora Ayala · Answer

undefined To determine if the circles overlap, compare the distances between the centers of the circles and the sum of their radii. If the distance between the centers is less than the sum of their radii, then the circles overlap. If it's equal, they touch at one point, and if it's greater, they do not overlap.

1. Calculate the distance between the centers of the circles using the 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 the coordinates of the centers of Circle A and Circle B, respectively.

2. Calculate the radii of the circles using the formula for the area of a circle:
   $$ \text{Area} = \pi r^2 $$
   Solve for $ r $ to find the radius of each circle.

3. Compare the distance between the centers ($ d $) with the sum of their radii ($ r_A + r_B $).

a. If $ d < r_A + r_B $, the circles overlap.
   b. If $ d = r_A + r_B $, the circles touch at one point.
   c. If $ d > r_A + r_B $, the circles do not overlap.

Performing these calculations will determine if the circles overlap.