Circle A has a center at #(1 ,5 )# and an area of #18 pi#. Circle B has a center at #(8 ,4 )# and an area of #66 pi#. Do the circles overlap?

Question

Circle A has a center at #(1   ,5  )# and an area of #18 pi#.  Circle B has a center at #(8  ,4  )# and an area of #66 pi#.  Do the circles overlap?

Caroline Aucoin · Accepted Answer

Get the distance between the centers.
For the circles to overlap, the distance between the centers should be less than or equal to the sum of the radii but greater than or equal to the difference #A_A = pir_A^2 = 18pi# #=> r_A^2 = 18# #=> r_A = 3sqrt2 ~~ 4.2...# #A_B = pir_B^2 = 66pi# #=> r_B^2 = 66# #=> r_B = sqrt66 ~~ 8...# Distance between centers: #D = sqrt((x_A - x_B)^2 + (y_A - y_B)^2)# #=> D = sqrt((1 - 8)^2 + (5 - 4)^2)# #=> D = sqrt((-7)^2 + 1^2)# #=> D = sqrt50# #=> D = 5sqrt2 ~~ 7...# #|r_A - r_B| ~~ 3.8# #r_A + r_B ~~ 12.2# The approximate distance between the centers is less than the sum of the radii but greater than the difference, the circles should overlap.

Sadie Adams · Answer

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

First, we need to find the radii of both circles. We know that the area of a circle ($A$) is given by the formula $A = \pi r^2$.

Given:
For Circle A: $A = 18\pi$
For Circle B: $A = 66\pi$

Using the formula for the area of a circle, we can find the radii of both circles:

For Circle A:
$$18\pi = \pi r_A^2$$
$$r_A^2 = \frac{18\pi}{\pi}$$
$$r_A^2 = 18$$
$$r_A = \sqrt{18}$$

For Circle B:
$$66\pi = \pi r_B^2$$
$$r_B^2 = \frac{66\pi}{\pi}$$
$$r_B^2 = 66$$
$$r_B = \sqrt{66}$$

Now, we find the distance between the centers of the circles using the distance formula:

Given:
Center of Circle A: (1, 5)
Center of Circle B: (8, 4)

$$ d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} $$

$$ d = \sqrt{(8 - 1)^2 + (4 - 5)^2} $$
$$ d = \sqrt{7^2 + (-1)^2} $$
$$ d = \sqrt{49 + 1} $$
$$ d = \sqrt{50} $$

Now, we compare the distance between the centers to the sum of the radii:

$$ \text{Sum of radii} = r_A + r_B = \sqrt{18} + \sqrt{66} $$

If the distance between the centers ($\sqrt{50}$) is less than the sum of the radii, the circles overlap.

If $\sqrt{50} < \sqrt{18} + \sqrt{66}$, then the circles overlap.

Since $\sqrt{50} = \sqrt{25 \times 2} = 5\sqrt{2}$, and $\sqrt{18} + \sqrt{66} > 5\sqrt{2}$, the circles do not overlap.