# A triangle has corners at #(3 ,7 )#, #(2 ,9 )#, and #(8 ,4 )#. What is the area of the triangle's circumscribed circle?

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If you just need the area of the circumscribed circle, the amount of calculations can be reduced a bit by appealing to a simple consequence of the sine law of triangles.

We usually express this law in the form

what is often not mentioned is that this common value is, actually the diameter

Here ABC is a triangle whose circumscribed circle is drawn in red. AD is a diameter of the circle. Then

Armed with this result, we first calculate the lengths of the three sides of the given triangle. Let us label the vertices

We also need one of the angles, say

With our numbers, we have

Finally

So, the area of the circumscribed circle is

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area

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Set point 1 as

Set point 2 as

Set point 3 as

The perpendicular from

(circumcentre).

Set mid point as

Set gradient

Perpendicular gradient

So we have

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Set gradient

Perpendicular gradient

So we have

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Using simultaneous equations for

the centre of the circle is at:

From this we can derive the radius and hence the area of the circle.

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Using Pythagoras on the two points

Let the radius be

area

area

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