# A triangle has corners at #(9 ,3 )#, #(4 ,1 )#, and #(2 ,4 )#. What is the area of the triangle's circumscribed circle?

The area of the triangle's circumscribed circle is

Let ,

Using Distance formula ,we get

Using cosine Formula ,we get

We know that,

Using sine formula:we get

So , the area of the triangle's circumscribed circle is:

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Note:

If we take,

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To find the area of the circumscribed circle of a triangle, we need to find the radius of the circumscribed circle first. The radius ( R ) can be calculated using the formula:

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

Where:

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

Then, once you have the radius ( R ), you can find the area of the circumscribed circle using the formula:

[ Area = \pi R^2 ]

For the given triangle with vertices at ( (9,3), (4,1), ) and ( (2,4) ):

- Find the side lengths of the triangle using the distance formula between the vertices.
- Use Heron's formula to find the area of the triangle.
- Calculate the radius using the formula above.
- Finally, find the area of the circumscribed circle using the formula above.

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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 #(5 ,4 )# and it passes through #(1 ,4 )#. What is the length of an arc covering #(5pi ) /8 # radians on the circle?
- Points #(4 ,4 )# and #(7 ,3 )# are #(5 pi)/4 # radians apart on a circle. What is the shortest arc length between the points?
- Two circles have the following equations: #(x +6 )^2+(y -5 )^2= 64 # and #(x -9 )^2+(y +4 )^2= 81 #. 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 circle's center is at #(2 ,4 )# and it passes through #(1 ,2 )#. What is the length of an arc covering #(5pi ) /4 # radians on the circle?
- What is the equation of the circle with a center at #(5 ,-3 )# and a radius of #6 #?

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