# A triangle has corners at #(8 ,1 )#, #(4 ,5 )#, and #(6 ,7 )#. What is the area of the triangle's circumscribed circle?

Area of triangle's circumscribed circle is

Area of triangle's circumscribed circle is

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Area of circumcircle

Refer figure abov.

B is mid point of AB

Coordinates of

Similarly coordinates of

Slope of AD

Slope of perpendicular line through B

Eqn of perpendicular line through B is

Slope of CD

Slope of perpendicular line through F

Eqn of perpendicular line through F is

Solving Eqns (1), (2) we get the coordinates of circumcenter O.

Radius of circumcircle is distance of O from the vertices A, D or C.

I.e

Area of circumcircle

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The area of the triangle's circumscribed circle can be calculated using 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:

( a = \sqrt{{(x_2 - x_1)^2 + (y_2 - y_1)^2}} )

( b = \sqrt{{(x_3 - x_2)^2 + (y_3 - y_2)^2}} )

( c = \sqrt{{(x_3 - x_1)^2 + (y_3 - y_1)^2}} )

Where ( (x_1, y_1) ), ( (x_2, y_2) ), and ( (x_3, y_3) ) are the coordinates of the triangle's vertices.

Then, calculate the semi-perimeter of the triangle:

( s = \frac{{a + b + c}}{2} )

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

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

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

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

Finally, use the formula for the area of the circle:

( A_{circle} = \pi R^2 )

Substitute the value of ( R ) into the formula 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.

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