The standard equation of a circle is:
(x - h)^2 + (y - k)^2 = r^2
where (h,k) is the centerpoint and r is the radius.
The circumscribed circle will include the three points of the triangle, therefore, we can use the 3 points to write 3 equations:
(6 - h)^2 + (8 - k)^2 = r^2 [1]
(7 - h)^2 + (5 - k)^2 = r^2 [2]
(3 - h)^2 + (9 - k)^2 = r^2 [3]
We can eliminate the variable r by setting the left side of equation [1] equal to the left side of equation [2] and setting the left side of equation [1] equal to the left side of equation [3]:
(6 - h)^2 + (8 - k)^2 = (7 - h)^2 + (5 - k)^2 [1 = 2]
(6 - h)^2 + (8 - k)^2 = (3 - h)^2 + (9 - k)^2 [1 = 3]
Expand the squares:
36 - 12h+ h^2 + 64 - 16k+ k^2 = 49 - 14h+ h^2 + 25 - 10k+ k^2
36 - 12h+ h^2 + 64 - 16k+ k^2 = 9 - 6h +h^2 + 81 - 18k + k^2
The square terms cancel:
36 - 12h+ cancelh^2 + 64 - 16k+ cancelk^2 = 49 - 14h+ cancelh^2 + 25 - 10k+ cancelk^2
36 - 12h+ cancelh^2 + 64 - 16k+ cancelk^2 = 9 - 6h + cancelh^2 + 81 - 18k + cancelk^2
36 - 12h + 64 - 16k = 49 - 14h + 25 - 10k
36 - 12h + 64 - 16k = 9 - 6h + 81 - 18k
Collect the constant terms and the terms containing h on the right and collect the terms containing k on the left:
-6k = -2h - 26 [4]
2k = 6h - 10 [5]
Multiply equation [5] by 3 and add to equation [4]:
0 = 16h - 56
h = 56/16 = 7/2
Find the value of k by substituting 7/2 for h into equation [5]:
2k = 6(7/2) - 10
k = 11/2
The centerpoint is (7/2, 11/2)
Find the value of r^2 by substituting the center point into equation [1]:
(6 - 7/2)^2 + (8 - 11/2)^2 = r^2
(12/2 - 7/2)^2 + (16 - 11/2)^2 = r^2
(5/2)^2 + (5/2)^2 = r^2
r^2 = 50/4 = 25/2
Multiply by pi, to find the area of the circle:
A = pir^2 = (25pi)/2
