# A triangle has corners at #(8 ,7 )#, #(2 ,1 )#, and #(5 ,6 )#. What is the area of the triangle's circumscribed circle?

The area of the circle is

Then,

The area of the circle is

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The area of the triangle's circumscribed circle can be calculated using the formula:

[A = \frac{abc}{4R}]

Where:

- (a), (b), and (c) are the side lengths of the triangle.
- (R) is the radius of the circumscribed circle.

First, calculate the side lengths of the triangle using the distance formula:

[a = \sqrt{(8 - 2)^2 + (7 - 1)^2}] [b = \sqrt{(2 - 5)^2 + (1 - 6)^2}] [c = \sqrt{(5 - 8)^2 + (6 - 7)^2}]

Then, calculate the semiperimeter of the triangle:

[s = \frac{a + b + c}{2}]

Next, compute the radius of the circumscribed circle using Heron's formula:

[A_{\text{triangle}} = \sqrt{s(s-a)(s-b)(s-c)}]

Finally, use the formula for the area of the circumscribed circle:

[A = \frac{abc}{4A_{\text{triangle}}}]

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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 ,7 )#, #(3 ,4 )#, and #(6 ,2 )#. What is the area of the triangle's circumscribed circle?
- Two circles have the following equations #(x -1 )^2+(y -4 )^2= 25 # and #(x +3 )^2+(y +3 )^2= 49 #. Does one circle contain the other? If not, what is the greatest possible distance between a point on one circle and another point on the other?
- A triangle has corners at #(2 , 9 )#, #(3 ,7 )#, and #(1 ,1 )#. What is the radius of the triangle's inscribed circle?
- Find the radius and center of a circle with the equation #x^2 + y^2 - 8x + 2y + 8 = 0# ?
- A circle has a chord that goes from #( pi)/2 # to #(15 pi) / 8 # radians on the circle. If the area of the circle is #121 pi #, what is the length of the chord?

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