Circle A has a center at #(12 ,2 )# and an area of #13 pi#. Circle B has a center at #(3 ,6 )# and an area of #28 pi#. Do the circles overlap?

Question

Circle A has a center at #(12   ,2  )# and an area of #13 pi#.  Circle B has a center at #(3  ,6   )# and an area of #28 pi#.  Do the circles overlap?

Sadie Allen · Accepted Answer

We can find the radius of each circle given its area. If the centers are further apart than the sum of the two radii, the circles don't overlap. If not, they do. The radii are 3.6 and 5.3 units, totaling 8.9 units, and the distance between the centers is 9.85 units, so the circles do not overlap.

The area of a circle is given by #A=pir^2#. Rearranging to make #r# the subject: #r = sqrt(A/pi)# Circle A: #r = sqrt(A/pi)=sqrt((13pi)/pi)=sqrt13~~3.6# Circle B: #r = sqrt(A/pi)=sqrt((28pi)/pi)=sqrt28~~5.3# The sum of these two radii is #8.9# units. The distance between the centers is given by: #d=sqrt((x_2-x_1)^2+(y_2-y_1)^2)# #=sqrt((12-3)^2+(6-2)^2)=sqrt(81+16)=sqrt97~~9.85# units. Since this is greater than the sum of the radii, the circles do not overlap.

Autumn Armstrong · Answer

undefined To determine if the circles overlap, we need to calculate the distance between their centers and compare it to the sum of their radii.

The distance between the centers of two circles with coordinates $(x_1, y_1)$ and $(x_2, y_2)$ is given by the distance formula:

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

For Circle A with center $(12, 2)$ and Circle B with center $(3, 6)$, the distance between their centers is:

$$ d = \sqrt{(3 - 12)^2 + (6 - 2)^2} = \sqrt{(-9)^2 + (4)^2} = \sqrt{81 + 16} = \sqrt{97} $$

Now, we need to compare this distance to the sum of the radii of the two circles. Since the area of a circle is $A = \pi r^2$, we can find the radii of the circles using the given areas:

For Circle A with area $13\pi$:

$$ 13\pi = \pi r_1^2 $$
$$ r_1^2 = 13 $$
$$ r_1 = \sqrt{13} $$

For Circle B with area $28\pi$:

$$ 28\pi = \pi r_2^2 $$
$$ r_2^2 = 28 $$
$$ r_2 = \sqrt{28} $$

The sum of the radii is $ r_1 + r_2 = \sqrt{13} + \sqrt{28} $.

If the distance between the centers is greater than the sum of the radii, the circles do not overlap. Otherwise, they overlap.

Comparing $ \sqrt{97} $ to $ \sqrt{13} + \sqrt{28} $ will determine if the circles overlap.