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Cetermine the length of AB - [Canadian Repêchage 2011]

Source:

January 15, 2011
trigonometrygeometry proposedgeometry

Problem Statement

In the diagram, the circle has radius 7\sqrt 7 and and centre O.O. Points A,BA, B and CC are on the circle. If BOC=120\angle BOC=120^\circ and AC=AB+1,AC = AB + 1, determine the length of AB.AB. [asy] import graph; size(120); real lsf = 0.5; pen dp = linewidth(0.7) + fontsize(10); defaultpen(dp); pen ds = black; pen qqttff = rgb(0,0.2,1); pen xdxdff = rgb(0.49,0.49,1); pen fftttt = rgb(1,0.2,0.2); draw(circle((2.34,2.4),2.01),qqttff); draw((2.34,2.4)--(1.09,0.82),fftttt); draw((2.34,2.4)--(4.1,1.41),fftttt); draw((1.09,0.82)--(1.4,4.18),fftttt); draw((4.1,1.41)--(1.4,4.18),fftttt); dot((2.34,2.4),ds); label("OO", (2.1,2.66),NE*lsf); dot((1.09,0.82),ds); label("BB", (0.86,0.46),NE*lsf); dot((4.1,1.41),ds); label("CC", (4.2,1.08),NE*lsf); dot((1.4,4.18),ds); label("AA", (1.22,4.48),NE*lsf); clip((-4.34,-10.94)--(-4.34,6.3)--(16.14,6.3)--(16.14,-10.94)--cycle); [/asy]