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let angleA, angleB and angleC " be " 2x,3x and 7x∠A,∠Band∠C be 2x,3xand7x, respectively,
=> 2x+3x+7x=12x=180^@⇒2x+3x+7x=12x=180∘,
=> x=15^@, => 2x=30^@, 3x=45^@, and 7x=105^@⇒x=15∘,⇒2x=30∘,3x=45∘,and7x=105∘,
as the shortest side is opposite the smallest angle,
=> BC=2012⇒BC=2012 cm
Let O and rOandr be the center and the radius of the circle, respectively.
Draw the diameter BODBOD, as shown in the figure.
=> BOD=2r, => angleBCD=90^@⇒BOD=2r,⇒∠BCD=90∘,
as angleBDC and angleBAC∠BDCand∠BAC subtend the same arc BCBC,
=> angleBDC=angleBAC=30^@⇒∠BDC=∠BAC=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.
