# A triangle has corners at #(3 ,5 )#, #(4 ,9 )#, and #(7 ,6 )#. What is the area of the triangle's circumscribed circle?

The area of the circumscribed circle is:

When I do this type problem, I always shift all of the corners so that one of them is the origin. This does not change the area of the circumscribed circle and it makes one of the 3 equations (that we must write) become very simple and useful.

Subtract 3 from every x coordinate and subtract 5 from every y coordinate:

Use the standard form of the equation of a circle,

, and the 3 shifted points to write three equations:

Please observe that equation [1] simplifies to:

Expand the squares, using the pattern #(a - b)^2 = a^2 - 2ab + b^2:

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

Multiply equation [8] by -4 and add to equation [7]:

Substitute 1.7 for h in equation [8]:

The area of a circle is:

The area of the circumscribed circle is:

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To find the area of the triangle's circumscribed circle, follow these steps:

- Calculate the lengths of the sides of the triangle using the distance formula.
- Use Heron's formula to find the area of the triangle.
- Find the radius of the circumscribed circle using the formula (R = \frac{abc}{4A}), where (a), (b), and (c) are the lengths of the sides of the triangle and (A) is its area.
- Finally, calculate the area of the circle using the formula for the area of a 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.

- A triangle has vertices A, B, and C. Vertex A has an angle of #pi/2 #, vertex B has an angle of #( 3pi)/8 #, and the triangle's area is #16 #. What is the area of the triangle's incircle?
- A triangle has corners at #(3 , 5 )#, #(4 ,2 )#, and #(8 ,4 )#. What is the radius of the triangle's inscribed circle?
- A triangle has corners at #(5 ,1 )#, #(7 ,9 )#, and #(4 ,3 )#. What is the area of the triangle's circumscribed circle?
- A circle has a center that falls on the line #y = 12/7x +3 # and passes through # ( 9 ,5 )# and #(8 ,7 )#. What is the equation of the circle?
- A triangle has vertices A, B, and C. Vertex A has an angle of #pi/12 #, vertex B has an angle of #pi/4 #, and the triangle's area is #3 #. What is the area of the triangle's incircle?

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