# Circle A has a center at #(1 ,8 )# and an area of #32 pi#. Circle B has a center at #(2 ,3 )# and an area of #28 pi#. Do the circles overlap?

The circles overlap.

Since the radius of Circle A is greater than the distance between the centers of the two circles, the center of Circle B is inside Circle A and the circles must overlap

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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 the radii, the circles overlap. If the distance between the centers is equal to the sum of the radii, the circles touch at one point. Otherwise, if the distance between the centers is greater than the sum of the radii, the circles do not overlap.

The distance between two points ((x_1, y_1)) and ((x_2, y_2)) is given by the formula:

[ \text{Distance} = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} ]

For Circle A with center ((1, 8)) and Circle B with center ((2, 3)):

[ \text{Distance} = \sqrt{(2 - 1)^2 + (3 - 8)^2} ] [ = \sqrt{1^2 + (-5)^2} ] [ = \sqrt{1 + 25} ] [ = \sqrt{26} ]

The radius of Circle A (denoted as (r_A)) is calculated from its area:

[ \text{Area}_A = \pi r_A^2 ] [ 32\pi = \pi r_A^2 ] [ r_A = \sqrt{32} = 4\sqrt{2} ]

The radius of Circle B (denoted as (r_B)) is calculated similarly:

[ \text{Area}_B = \pi r_B^2 ] [ 28\pi = \pi r_B^2 ] [ r_B = \sqrt{28} = 2\sqrt{7} ]

The sum of the radii ((r_A + r_B)) is:

[ 4\sqrt{2} + 2\sqrt{7} ]

Comparing the distance between the centers ((\sqrt{26})) to the sum of the radii, if (\sqrt{26}) is less than (4\sqrt{2} + 2\sqrt{7}), the circles overlap. Otherwise, they do not overlap.

Since (\sqrt{26} < 4\sqrt{2} + 2\sqrt{7}), the circles overlap.

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When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

- A line passes through #(3 ,2 )# and #(1 ,3 )#. A second line passes through #(1 , 4 )#. What is one other point that the second line may pass through if it is parallel to the first line?
- An isosceles triangle has sides A, B, and C with sides B and C being equal in length. If side A goes from #(7 ,5 )# to #(8 ,2 )# and the triangle's area is #36 #, what are the possible coordinates of the triangle's third corner?
- A triangle has corners at #(2 ,9 )#, #(4 ,8 )#, and #(5 ,1 )#. How far is the triangle's centroid from the origin?
- A triangle has corners at #(7 ,9 )#, #(4 ,1 )#, and #(3 ,8 )#. How far is the triangle's centroid from the origin?
- A line passes through #(5 ,9 )# and #(8 ,3 )#. A second line passes through #(2 ,6 )#. What is one other point that the second line may pass through if it is parallel to the first line?

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