A triangle has corners at #(4 ,7 )#, #(3 ,4 )#, and #(8 ,9 )#. What is the area of the triangle's circumscribed circle?
2 Answers
The area of the circumscribed circle is
Explanation:
To calculate the area of the circle, we must calculate the radius
Let the center of the circle be
Then,
We have
From
From
From equations
The center of the circle is
The area of the circle is
Explanation:
Here's the shortcut.
The circumcircle is just the circle through the three vertices; the triangle almost doesn't matter. Except, miraculously, the circumradius
It's much more useful squared, and we're looking for
The coordinates give the squared distances easily. Archimedes' Theorem relates the squared distances to the triangle area:
So,
We form the squared distances from pairs of points