The circle circumscribed on a triangle is the one that passes through the three vertices.
The general equation for a circle with center (a,b)(a,b) and squared radius kk is
(x - a)^2 + (y-b)^2 = k (x−a)2+(y−b)2=k
Substituting our three points,
(8 - a)^2 + (3 - b)^2 = k (8−a)2+(3−b)2=k
(2 - a)^2 + (4- b)^2 = k(2−a)2+(4−b)2=k
(7 -a )^2 + (2-b)^2 = k(7−a)2+(2−b)2=k
a^2 + b^2 - 16 a - 6 b + 64 + 9 = k a2+b2−16a−6b+64+9=k
a^2 + b^2 - 4a - 8b + 4 + 16 = k a2+b2−4a−8b+4+16=k
a^2 + b^2 - 14 a - 4 b + 49 + 4 = ka2+b2−14a−4b+49+4=k
Subtracting pairs,
-12 a + 2b + 53 = 0−12a+2b+53=0
-2 a -2 b + 20 = 0−2a−2b+20=0
Adding,
-14 a + 73 = 0 −14a+73=0
a = 73/14a=7314
12 a + 12 b - 120 = 0 12a+12b−120=0
14b - 67 = 014b−67=0
b = 67/14b=6714
k = (2 - 73/14 )^2 + (4- 67/14)^2 k=(2−7314)2+(4−6714)2
k = 1/14^2 ( (28-73 )^2 + (56-67)^2 ) = 2146/14^2 = 1073/98 k=1142((28−73)2+(56−67)2)=2146142=107398
Usually they pick nicer numbers for these problems. Our circle's area is A = \pi r^2 = \pi kA=πr2=πk
A = frac { 1073 pi }{98} A=1073π98
