# 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?

The circles will intersect at two points.

Two circles intersect if, and only if, the distance between their

centers is between the sum and the difference of their radii.

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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.

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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.

- Circle A has a center at #(2 ,4 )# and an area of #81 pi#. Circle B has a center at #(4 ,3 )# and an area of #36 pi#. Do the circles overlap? If not, what is the shortest distance between them?
- An isosceles triangle has sides A, B, and C with sides B and C being equal in length. If side A goes from #(2 ,5 )# to #(8 ,1 )# and the triangle's area is #15 #, what are the possible coordinates of the triangle's third corner?
- What is the perimeter of a triangle with corners at #(1 ,5 )#, #(8 ,3 )#, and #(4 ,1 )#?
- Circle A has a center at #(2 ,8 )# and a radius of #2 #. Circle B has a center at #(-8 ,3 )# and a radius of #1 #. Do the circles overlap? If not, what is the smallest distance between them?
- A triangle has corners at #(3 ,5 )#, #(4 ,7 )#, and #(1 ,2 )#. How far is the triangle's centroid from the origin?

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