Circle A has a center at #(3 ,4 )# and an area of #64 pi#. Circle B has a center at #(1 ,12 )# and an area of #54 pi#. Do the circles overlap?

Question

Circle A has a center at #(3   ,4  )# and an area of #64 pi#.  Circle B has a center at #(1  ,12   )# and an area of #54 pi#.  Do the circles overlap?

Savannah Aguilar · Accepted Answer

The circles will intersect at two points.

Area of circle #A# is #A_A=pi *r_A^2= 64 pi :. r_A=8# Area of circle #B# is #A_B=pi *r_B^2= 54 pi # # :. r_B=sqrt 54~~7.35 (2dp) :. r_A+r_B= 15.35 # and # |r_A-r_B| = 0.65# Center of first circle #A# is at #(3,4)# and radius is #8# unit . Center of second circle #B# is at #(1,12)# and radius is #sqrt 54# unit . Distance between their centres is #d=sqrt((x_1-x_2)^2+(y_1-y_2)^2)=sqrt((3-1)^2+(4-12)^2) # or #d=sqrt(4+64)=sqrt 68 ~~ 8.25# unit. Two circles intersect if, and only if, the distance between their centers is between the sum and the difference of their radii. Here #0.65 < 8.25 <15.35#, so they intersect at two points. [Ans]

Jayden Jackson · Answer

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

First, calculate the radii of the circles using the formula: $ \text{Area} = \pi \times \text{radius}^2 $.

For Circle A: 
$ 64\pi = \pi \times \text{radius}_A^2 $
$ \text{radius}_A^2 = 64 $
$ \text{radius}_A = 8 $

For Circle B:
$ 54\pi = \pi \times \text{radius}_B^2 $
$ \text{radius}_B^2 = 54 $
$ \text{radius}_B = \sqrt{54} $

Next, calculate the distance between the centers using the distance formula:
$ \text{Distance} = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} $.

For Circle A and Circle B:
$ \text{Distance} = \sqrt{(1 - 3)^2 + (12 - 4)^2} $
$ \text{Distance} = \sqrt{(-2)^2 + (8)^2} $
$ \text{Distance} = \sqrt{4 + 64} $
$ \text{Distance} = \sqrt{68} $

Now, compare the distance between the centers to the sum of their radii:
$ \text{Distance} = \sqrt{68} \approx 8.246 $
$ \text{Sum of radii} = 8 + \sqrt{54} \approx 8 + 7.348 \approx 15.348 $

Since the distance between the centers ($ \sqrt{68} $) is greater than the sum of their radii ($ 8 + \sqrt{54} $), the circles do not overlap.