# Circle A has a center at #(1 ,3 )# and an area of #18 pi#. Circle B has a center at #(11 ,7 )# and an area of #54 pi#. Do the circles overlap?

Yes, the circles do overlap.

Circle A, center (1,3),

Circle B, center (11,7),

distance between the two center points

As the distance between the two center points is smaller than the sum of the two radii

Here's an image of the two circles

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To determine if the circles overlap, we first need to find the radii of each circle using the formula for the area of a circle:

[ \text{Area} = \pi \times \text{radius}^2 ]

For Circle A: [ 18\pi = \pi \times \text{radius}_A^2 ] [ \text{radius}_A^2 = \frac{18\pi}{\pi} ] [ \text{radius}_A^2 = 18 ] [ \text{radius}_A = \sqrt{18} ]

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

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

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

For Circle A with center (1, 3) and Circle B with center (11, 7): [ \text{Distance} = \sqrt{(11 - 1)^2 + (7 - 3)^2} ] [ \text{Distance} = \sqrt{10^2 + 4^2} ] [ \text{Distance} = \sqrt{100 + 16} ] [ \text{Distance} = \sqrt{116} ]

Since the radii of the circles are ( \sqrt{18} ) and ( \sqrt{54} ) respectively, and the distance between the centers of the circles is ( \sqrt{116} ), we can compare the sum of the radii with the distance between the centers:

If ( \text{radius}_A + \text{radius}_B < \text{Distance} ), the circles do not overlap.

If ( \text{radius}_A + \text{radius}_B = \text{Distance} ), the circles touch each other at exactly one point.

If ( \text{radius}_A + \text{radius}_B > \text{Distance} ), the circles overlap.

We calculate: [ \sqrt{18} + \sqrt{54} = \sqrt{18} + \sqrt{54} < \sqrt{116} ]

Therefore, 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 triangle has corners at #(4 ,4 )#, #(8 ,6 )#, and #(7 ,1 )#. How far is the triangle's centroid from the origin?
- A line passes through #(4 ,9 )# and #(1 ,6 )#. A second line passes through #(3 ,7 )#. What is one other point that the second line may pass through if it is parallel to the first line?
- A triangle has corners at #(6 ,7 )#, #(2 ,6 )#, and #(1 ,2 )#. How far is the triangle's centroid from the origin?
- Circle A has a center at #(7 ,5 )# and an area of #92 pi#. Circle B has a center at #(9 ,2 )# and an area of #14 pi#. Do the circles overlap?
- An isosceles triangle has sides #a,# #b,# and #c# with sides #a# and #c# being equal in length. If side #b# goes from #(5 ,1 )# to #(3 ,2 )# and the triangle's area is #8 #, what are the possible coordinates of the triangle's third vertex?

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