# Circle A has a center at #(11 ,2 )# and an area of #100 pi#. Circle B has a center at #(7 ,9 )# and an area of #36 pi#. Do the circles overlap? If not, what is the shortest distance between them?

The circles overlap.

The sum of the radii is

Therefore,

the circles overlap

graph{((x-11)^2+(y-2)^2-100)((x-7)^2+(y-9)^2-36)=0 [-20.64, 30.66, -8, 17.66]}

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To determine if the circles overlap, calculate the distance between their centers and compare it to the sum of their radii. If the distance between the centers is greater than the sum of their radii, the circles do not overlap. Otherwise, they do overlap.

The distance between the centers of Circle A and Circle B can be found using the distance formula:

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

Given the coordinates of the centers of the circles: [ (x_1, y_1) = (11, 2) ] [ (x_2, y_2) = (7, 9) ]

The distance between the centers is: [ \text{Distance} = \sqrt{(7 - 11)^2 + (9 - 2)^2} = \sqrt{16 + 49} = \sqrt{65} ]

The radius of Circle A can be found by taking the square root of its area: [ \text{Radius of Circle A} = \sqrt{\text{Area of Circle A} / \pi} = \sqrt{100} = 10 ]

The radius of Circle B can be found similarly: [ \text{Radius of Circle B} = \sqrt{\text{Area of Circle B} / \pi} = \sqrt{36} = 6 ]

The sum of their radii is (10 + 6 = 16).

Comparing the distance between the centers ((\sqrt{65})) to the sum of their radii (16), since ( \sqrt{65} > 16), the circles do not overlap.

To find the shortest distance between them, subtract the sum of their radii from the distance between their centers: [ \text{Shortest distance} = \sqrt{65} - 16 ]

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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.

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- A triangle has corners at #(4 ,1 )#, #(8 ,3 )#, and #(5 ,8 )#. How far is the triangle's centroid from the origin?
- What is the midpoint of the line segment joining the points (7, 4) and (-8, 7)?
- 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 #27 #, what are the possible coordinates of the triangle's third corner?
- What is the perimeter of a triangle with corners at #(3 ,4 )#, #(6 ,7 )#, and #(4 ,5 )#?

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