Using the 3 points and the standard equation of a circle, #r^2 = (x - h)^2 + (x - k)^2#, we can write 3 equations:
#r^2 = (3 - h)^2 + (5 - k)^2#
#r^2 = (7 - h)^2 + (9 - k)^2#
#r^2 = (4 - h)^2 + (2 - k)^2#
Because #r^2 = r^2#, we can set the right side of the first equation equal to the right side of the second equation and the third equation:
#(3 - h)^2 + (5 - k)^2 = (7 - h)^2 + (9 - k)^2#
#(3 - h)^2 + (5 - k)^2 = (4 - h)^2 + (2 - k)^2#
Expand the squares:
#9 - 6h + h^2 + 25 - 10k + k^2 = 49 - 14h + h^2 + 81 - 18k + k^2#
#9 - 6h + h^2 + 25 - 10k + k^2 = 16 - 8h + h^2 + 4 - 4k + k^2#
combine like terms and cancel the #h^2# and #k^2# terms:
#8k = 96 - 8h#
#-6k = -2h - 14#
Divide the first equation by 8:
#k = 12 - h#
#-6k = -2h - 14#
Substitution:
#-6(12 - h) = -2h - 14#
#-72 + 6h = -2h - 14#
#8h = 58#
#h = 29/4#
#k = 48/4 - 29/4#
#k = 19/4#
Substitute #(29/4, 19/4)# for #(h, k)#:
#r^2 = (3 - 29/4)^2 + (5 - 19/4)^2#
#r^2 = (-17/4)^2 + (1/4)^2#
#r^2 = 290/16 = 145/8#
The area of a circle is #pir^2#
#A = (145pi)/8#
