let #angleA, angleB and angleC " be " 2x,3x and 7x#, respectively,
#=> 2x+3x+7x=12x=180^@#,
#=> x=15^@, => 2x=30^@, 3x=45^@, and 7x=105^@#,
as the shortest side is opposite the smallest angle,
#=> BC=2012# cm
Let #O and r# be the center and the radius of the circle, respectively.
Draw the diameter #BOD#, as shown in the figure.
#=> BOD=2r, => angleBCD=90^@#,
as #angleBDC and angleBAC# subtend the same arc #BC#,
#=> angleBDC=angleBAC=30^@#
In #DeltaBDC, sin30=(BC)/(BD)=2012/(2r)#
#=> r=2012/(2*sin30)=2012/(2*1/2)=2012# cm
Footnotes : if you know one side of a triangle and its opposite angle, the radius of the circumcircle is given by : #r=1/2*a/sinA#, where #a# is the length of one side and #A# is the angle opposite that side.
