# A triangle has corners at #(4 ,6 )#, #(2 ,9 )#, and #(8 ,4 )#. What is the area of the triangle's circumscribed circle?

Area of circumscribed circle is

And radius of circumscribed circle is

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To find the area of the circumscribed circle of a triangle, we first need to find the lengths of the sides of the triangle. Then, we can use these side lengths to calculate the radius of the circumscribed circle. Once we have the radius, we can use the formula for the area of a circle to find the area of the circumscribed circle.

Using the given coordinates, we can calculate the lengths of the sides of the triangle using the distance formula:

Side 1: ( \sqrt{(4 - 2)^2 + (6 - 9)^2} = \sqrt{4 + 9} = \sqrt{13} ) Side 2: ( \sqrt{(4 - 8)^2 + (6 - 4)^2} = \sqrt{16 + 4} = \sqrt{20} = 2\sqrt{5} ) Side 3: ( \sqrt{(2 - 8)^2 + (9 - 4)^2} = \sqrt{36 + 25} = \sqrt{61} )

Next, we can use these side lengths to find the radius of the circumscribed circle using the formula:

( R = \frac{abc}{4A} )

Where ( a ), ( b ), and ( c ) are the lengths of the sides of the triangle, and ( A ) is the area of the triangle.

Using Heron's formula to find the area of the triangle:

( s = \frac{a + b + c}{2} = \frac{\sqrt{13} + 2\sqrt{5} + \sqrt{61}}{2} )

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

Finally, once we have the radius, we can use the formula for the area of a circle to find the area of the circumscribed circle:

( Area = \pi R^2 )

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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 circle's center is at #(8 ,7 )# and it passes through #(6 ,2 )#. What is the length of an arc covering #(5 pi ) /6 # radians on the circle?
- A circle's center is at #(9 ,1 )# and it passes through #(5 ,4 )#. What is the length of an arc covering #(pi ) /3 # radians on the circle?
- A circle has a center that falls on the line #y = 2x +7 # and passes through #(4 ,4 )# and #(1 ,2 )#. What is the equation of the circle?
- A triangle has corners at #(5 ,3 )#, #(2 ,4 )#, and #(7 ,2 )#. What is the area of the triangle's circumscribed circle?
- A circle's center is at #(2 ,6 )# and it passes through #(3 ,8 )#. What is the length of an arc covering #(2pi ) /3 # radians on the circle?

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