If the sides of a triangle are #a#, #b# and #c#, then the area of the triangle #Delta# is given by the formula
#Delta=sqrt(s(s-a)(s-b)(s-c))#, where #s=1/2(a+b+c)#
and radius of circumscribed circle is #(abc)/(4Delta)#
Hence let us find the sides of triangle formed by #(4,6)#, #(2,9)# and #(8,4)#. This will be surely distance between pair of points, which is
#a=sqrt((2-4)^2+(9-6)^2)=sqrt(4+9)=sqrt13=3.6056#
#b=sqrt((8-2)^2+(4-9)^2)=sqrt(36+25)=sqrt61=7.8102# and
#c=sqrt((8-4)^2+(4-6)^2)=sqrt(16+4)=sqrt20=4.4721#
Hence #s=1/2(3.6056+7.8102+4.4721)=1/2xx15.8879=7.944#
and #Delta=sqrt(7.944xx(7.944-3.6056)xx(7.944-7.8102)xx(7.944-4.4721)#
= #sqrt(7.944xx4.3384xx0.1338xx3.4719)=sqrt16.01=4.0013#
And radius of circumscribed circle is
#(3.6056xx7.8102xx4.4721)/(4xx4.0013)=7.8685#
And area of circumscribed circle is #3.1416xx(7.8685)^2=194.5068#
