The standard Cartesian form of the equation of a circle is:
(x - h)^2 + (y - k)^2 = r^2" [1]"
where (x, y) is any point on the circle, (h, k) is the center point and r is the radius.
Use equation [1] and the 3 points to write 3 equations:
(3 - h)^2 + (8 - k)^2 = r^2" [2]"
(7 - h)^2 + (9 - k)^2 = r^2" [3]"
(4 - h)^2 + (6 - k)^2 = r^2" [4]"
Expand the squares, using the pattern (a - b) = a^2 - 2ab + b^2:
9 - 6h + h^2 + 64 - 16k + k^2 = r^2" [5]"
49 - 14h + h^2 + 81 - 18k + k^2 = r^2" [6]"
16 - 8h + h^2 + 36 - 12k + k^2 = r^2" [7]"
Set the left side of equation [5] equal to the left side of equation [6]:
9 - 6h + h^2 + 64 - 16k + k^2 = 49 - 14h + h^2 + 81 - 18k + k^2" [8]"
Set the left side of equation [7] equal to the left side of equation [6]:
16 - 8h + h^2 + 36 - 12k + k^2 = 49 - 14h + h^2 + 81 - 18k + k^2" [9]"
There h^2 and k^2 terms on both sides of equations [8] and [9], therefore, they sum to zero:
9 - 6h + 64 - 16k = 49 - 14h + 81 - 18k" [10]"
16 - 8h + 36 - 12k = 49 - 14h + 81 - 18k" [11]"
Collect all of the constant terms into a single term on the right:
- 6h - 16k = - 14h - 18k + 57" [12]"
- 8h - 12k = - 14h - 18k + 78" [13]"
Collect the h terms into a single term on the left:
8h - 16k = - 18k + 57" [14]"
6h - 12k = - 18k + 78" [15]"
Collect the k terms into a single term on the left:
8h + 2k = 57" [16]"
6h + 6k = 78" [17]"
Multiply equation [17] by -1/3 and add to equation [16]:
6h = 31
h = 31/6
Substitute 31/6 for h into equation [17]:
6(31/6) + 6k = 78" [17]"
k = 47/6
Substitute the values for h and k into equation [3}:
(7 - 31/6)^2 + (9 - 47/6)^2 = r^2
r^2 = (42/6 - 31/6)^2 + (54 - 47/6)^2
r^2 = 170/36
Area = pir^2
Area = (170pi)/36
