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#
