# Circle A has a center at #(3 ,5 )# and an area of #78 pi#. Circle B has a center at #(1 ,2 )# and an area of #54 pi#. Do the circles overlap?

Yes

Proof: graph{((x-3)^2+(y-5)^2-54)((x-1)^2+(y-2)^2-78)=0 [-20.33, 19.67, -7.36, 12.64]}

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These overlap if

We can skip the calculator and check

Our radii are squared.

and the squared separation of the centers

It's pretty crazy that we all automatically grab a calculator or computer and start taking square roots when the squared lengths are all nice integers.

That's already superior to Heron. Nevertheless, let's move on. I'll cut out the boring parts.

As one might anticipate from an area formula, that is nicely symmetric. Let's make it appear less symmetrical.

Adding,

This formula determines a triangle's squared area based on the squared lengths of its sides; if the latter are rational, the former is also.

Non-overlapping circles, or an imaginary area if we had received a negative value, indicate that the triangle is not real.

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To determine if the circles overlap, we need to compare the distances 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.

The distance between two points ((x_1, y_1)) and ((x_2, y_2)) is given by the formula: [ d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} ]

The radius (r) of a circle with area (A) is given by: [ r = \sqrt{\frac{A}{\pi}} ]

Using the given information, we can find the radii of Circle A and Circle B using their respective areas. Then, we can calculate the distance between their centers using the given coordinates. Finally, we compare this distance to the sum of the radii to determine if the circles overlap. If the distance is less than the sum of the radii, the circles overlap; otherwise, they 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.

- What is the perimeter of a triangle with corners at #(7 ,5 )#, #(1 ,2 )#, and #(4 ,8 )#?
- Circle A has a center at #(9 ,-2 )# and a radius of #1 #. Circle B has a center at #(1 ,3 )# and a radius of #3 #. Do the circles overlap? If not, what is the smallest distance between them?
- Circle A has a center at #(3 ,7 )# and an area of #56 pi#. Circle B has a center at #(9 ,6 )# and an area of #28 pi#. Do the circles overlap?
- Circle A has a center at #(6 ,2 )# and an area of #45 pi#. Circle B has a center at #(2 ,3 )# and an area of #75 pi#. Do the circles overlap?
- Circle A has a center at #(2 ,3 )# and a radius of #5 #. Circle B has a center at #(3 ,8 )# and a radius of #1 #. Do the circles overlap? If not what is the smallest distance between them?

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