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 #(4,7)#, #(3,4)# and #(6,9)#. This will be surely distance between pair of points, which is
#a=sqrt((3-4)^2+(4-7)^2)=sqrt(1+9)=sqrt10=3.1623#
#b=sqrt((6-3)^2+(9-4)^2)=sqrt(9+25)=sqrt34=5.831# and
#c=sqrt((6-4)^2+(9-7)^2)=sqrt(4+4)=sqrt8=2.8284#
Hence #s=1/2(3.1623+5.831+2.8284)=1/2xx11.8217=5.9109#
and #Delta=sqrt(5.9109xx(5.9109-3.1623)xx(5.9109-5.831)xx(5.9109-2.8284))#
= #sqrt(5.9109xx2.7486xx0.0799xx3.0825)=sqrt4.0014=2.0004#
And radius of circumscribed circle is
#(3.1623xx5.831xx2.8284)/(4xx2.0004)=6.518#
And area of circumscribed circle is #3.1416xx(6.518)^2=133.47#
