# A triangle has corners at #(9 ,5 )#, #(2 ,1 )#, and #(3 ,6 )#. What is the area of the triangle's circumscribed circle?

Shift ALL the points so that one is the origin. Use the standard Cartesian form for the equation of a circle and the new points to write 3 equations. Use the 3 equations to solve for

Shift all 3 points so that one of them is the origin:

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

Use the new points to write 3 equations:

Expand the squares:

Subtract equation [6] from equation [5] and equation [7] from equation [5]:

Collect the constant terms into a single term on the right:

Multiply equation [11] by -7 and add to equation [10]:

Substitute the value for k into equation [11] and solve for h:

The area of the circle is:

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To find the area of the triangle's circumscribed circle, 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 sides of the triangle, and ( R ) is the radius of the circumscribed circle.

First, you need to calculate the lengths of the sides of the triangle using the given coordinates:

- Side ( a ) is the distance between (9, 5) and (2, 1).
- Side ( b ) is the distance between (2, 1) and (3, 6).
- Side ( c ) is the distance between (3, 6) and (9, 5).

Next, find the semi-perimeter of the triangle, ( s = \frac{a + b + c}{2} ).

Then, use Heron's formula to calculate the area of the triangle:

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

After finding the area of the triangle, you can determine the radius of the circumscribed circle using the formula ( R = \frac{abc}{4A} ).

Finally, calculate the area of the circumscribed circle using the formula for the area of a circle ( A_{circle} = \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.

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- Two circles have the following equations #(x +5 )^2+(y -2 )^2= 36 # and #(x +2 )^2+(y -1 )^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?
- Given: circle O with diameter CD C(-5,4) D(3,-2) create a equation if the circle?
- A circle's center is at #(7 ,5 )# and it passes through #(5 ,8 )#. What is the length of an arc covering #(5pi ) /3 # radians on the circle?

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