# A triangle has corners at #(7 ,4 )#, #(4 ,6 )#, and #(6 ,7 )#. What is the area of the triangle's circumscribed circle?

Let's begin by moving the triangle to the left 4 and down 6:

this does not change the size of the triangle or the circumscribed circle.

The standard form for the equation of a circle is:

Use the standard form and the 3 points to write 3 equations:

Collect the constant terms on right:

Multiply equation [9] by 2 and add to equation [8]:

h = 23/14

Substitute into equation [9]

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To find the area of the triangle's circumscribed circle, we first need to find the circumradius, which is the radius of the circle that passes through all three vertices of the triangle.

We can use the formula for the circumradius of a triangle, which is given by:

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

Where:

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

We can find the lengths of the sides of the triangle using the distance formula between the given points. Once we have the lengths of the sides, we can use Heron's formula to find the area of the triangle. Finally, we can plug the values into the formula for the circumradius to find the radius of the circumscribed circle.

After finding the radius of the circumscribed circle, we can use the formula for the area of a circle:

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

Substitute the value of the radius to find the area of the circumscribed circle.

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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 #(8 , 6 )#, #(4 ,3 )#, and #(1 ,4 )#. What is the radius of the triangle's inscribed circle?
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- A triangle has sides with lengths of 5, 8, and 3. What is the radius of the triangles inscribed circle?

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