# A triangle has corners at #(9 ,5 )#, #(2 ,3 )#, and #(7 ,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 [3.1] from equation [2.1]:

Combine like terms:

Subtract equation [3.1] from equation [4.1]:

Combine like terms:

#72 - 10h -6k - 3/2(93 - 14h - 4k) = 0

#72 - 10h -6k - 279/2 + 21h + 6k = 0

Combine like terms:

Use equation [6] to find the value of k:

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Circum center (6.1364, 1.7727)

Area of circumcircle 58.5065

Slope of AB m1 = (3-5)/(2-9)=2/7.

Slope of perpendicular at mid point of AB = -1/m1 = -7/2

Midpoint of AB = (9+2)/2, (3+5)/2 = 11/2, 4

Eqn of perpendicular bisector of AB is

Slope of BC m2 = (6-3)/(7-2) = 3/5.

Slope of perpendicular at mid point of AB = -1/m1 = -5/3.

Midpoint of BC = (7+2)/2, (6+3)/2 = 9/2, 9/2

Eqn of perpendicular bisector of BC is

Solving Eqns (1), (2)

Circum center (6.1364, 1.7727)

Area of circumcircle

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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 circumscribed circle, ( a ), ( b ), and ( c ) are the lengths of the sides of the triangle, and ( R ) is the radius of the circumscribed circle.

First, calculate the lengths of the sides of the triangle using the distance formula:

( d_{1} = \sqrt{{(x_{2} - x_{1})^2 + (y_{2} - y_{1})^2}} )

( d_{2} = \sqrt{{(x_{3} - x_{2})^2 + (y_{3} - y_{2})^2}} )

( d_{3} = \sqrt{{(x_{3} - x_{1})^2 + (y_{3} - y_{1})^2}} )

Then, calculate the semiperimeter (( s )) of the triangle:

( s = \frac{{d_{1} + d_{2} + d_{3}}}{2} )

Next, use Heron's formula to find the area of the triangle:

( A_{triangle} = \sqrt{{s(s - d_{1})(s - d_{2})(s - d_{3})}} )

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

( R = \frac{{abc}}{{4A_{triangle}}} )

Then, use the formula for the area of the circumscribed circle:

( A = \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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- The circumference of a circle measures #22pi# units. How do you find the area of the circle?
- A circle has a chord that goes from #( 3 pi)/4 # to #(5 pi) / 4 # radians on the circle. If the area of the circle is #16 pi #, what is the length of the chord?

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