# A triangle has corners at #(6 ,4 )#, #(8 ,2 )#, and #(3 ,6 )#. What is the area of the triangle's circumscribed circle?

The standard Cartesian form for the equation of a circle is:

Expand the squares:

Subtract equation [4.1] from equation [2.1]:

Subtract equation [4.1] from equation [3.1]:

Multiply equation [5] by -2 and add it to equation [6]:

The area of the circle is:

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To find the area of the circumscribed circle of a triangle, you can use the formula:

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

Where ( A ) is the area of the triangle, ( a ), ( b ), and ( c ) are the lengths of the triangle's sides, and ( R ) is the radius of the circumscribed circle.

First, you need to find the lengths of the sides of the triangle using the distance formula:

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

Once you have the lengths of the sides, you can then use Heron's formula to find the area of the triangle:

[ A = \sqrt{s(s - a)(s - b)(s - c)} ]

Where ( s ) is the semi-perimeter of the triangle, given by ( s = \frac{a + b + c}{2} ).

After finding the area of the triangle, you can calculate the radius of the circumscribed circle using the formula:

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

Finally, you can use the radius ( R ) to find the area of the circumscribed circle using the formula for the area of a circle:

[ A_{circle} = \pi R^2 ]

By plugging in the values you've calculated, you can find the area of the circumscribed circle of the 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.

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- Sarah has a triangular piece of paper. The base of the paper is 14 and 1/2 inches. The height of the paper is 12 inches. What is the area of the piece of paper?
- A circle's center is at #(7 ,2 )# and it passes through #(5 ,8 )#. What is the length of an arc covering #(7pi ) /4 # radians on the circle?
- A triangle has corners at #(4 ,6 )#, #(5 ,9 )#, and #(7 ,5 )#. What is the area of the triangle's circumscribed circle?

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