The standard form of the equation of circle is:
(x - h)^2 + (y - k)^2 = r^2
where (x,y) any point on the circle, (h, k) is the center, and r is the radius.
Move all three points to the left 3 and down 8:
A = (0,0), B = (2,1), and C = (1, -2)
This does not change the size of the triangle or the circumscribed circle but it greatly simplifies the first equation that we write, using the points:
h^2 + k^2 = r^2" [1]"
(2 - h)^2 + (1 - k)^2 = r^2" [2]"
(1 - h)^2 + (-2 - k)^2 = r^2" [3]"
Substitute h^2 + k^2 for r^2 equations [2] and [3]:
(2 - h)^2 + (1 - k)^2 = h^2 + k^2" [4]"
(1 - h)^2 + (-2 - k)^2 = h^2 + k^2" [5]"
Expand the squares:
4 - 4h + h^2 + 1 - 2k + k^2 = h^2 + k^2" [6]"
1 - 2h + h^2 + 4 + 4k + k^2 = h^2 + k^2" [7]"
The square terms cancel:
4 - 4h + cancel(h^2) + 1 - 2k + cancel(k^2) = cancel(h^2) + cancel(k^2)" [8]"
1 - 2h + cancel(h^2) + 4 + 4k + cancel(k^2) = cancel(h^2) + cancel(k^2)" [9]"
Collect the constant terms on the right:
-4h - 2k = -5" [10]"
-2h + 4k = -5" [11]"
Multiply equation [10] by 2 and add to equation [11]:
-10h = -15
Solve for h:
h = 3/2
Substitute 3/2 for h in equation [11]:
-2(3/2) + 4k = -5
4k = -2
k = -1/2
Use equation [1] to find the value of r^2
r^2 = (3/2)^2 + (-1/2)^2
r^2 = 10/4
The area of a circle is:
Area = pir^2
Substitute 10/4 for r^2:
Area = (10pi)/4
