The standard Cartesian form for 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.
The points #(6,4)#, #(8,2)#, and #(3,6)# must lie on the circumscribed circle, therefore, we can use these points to write 3 unique equations:
#(6-h)^2+(4-k)^2=r^2" [2]"#
#(8-h)^2+(2-k)^2=r^2" [3]"#
#(3-h)^2+(6-k)^2=r^2" [4]"#
Expand the squares:
#36-12h+h^2+16-8k+k^2=r^2" [2.1]"#
#64-16h+h^2+4-4k+k^2=r^2" [3.1]"#
#9-6h+h^2+36-12k+k^2=r^2" [4.1]"#
Subtract equation [4.1] from equation [2.1]:
#7-6h+4k=0" [5]"#
Subtract equation [4.1] from equation [3.1]:
#23-10h+8k=0" [6]"#
Multiply equation [5] by -2 and add it to equation [6]:
#9+2h=0#
#h = -9/2#
Substitute #h = -9/2# into equation [5] and then solve for #k#:
#7-6(-9/2)+4k=0#
#34 + 4k=0#
#k = -17/2#
Substitute #h = -9/2# and #k = -17/2# into equation [2] and the solve for #r^2#:
#(6+9/2)^2+(4+17/2)^2=r^2#
#(12/2+9/2)^2+(8/2+17/2)^2=r^2#
#(21/2)^2+(25/2)^2 = r^2#
#r^2 = 533/2#
The area of the circle is:
#"Area" = pir^2#
#"Area" = 533/2pi#
