+62 Since \(\triangle ABC\) is a 30-60-90 triangle and \(\overline{AB}\) is 1, \(\overline{AC}\) is \(\sqrt{3}\) and \(\overline{BC}\) is 2. Let the intersection of the ray \(\overline{AG}\) through \(\overline{BC}\) be D. Becuase \(\overline{AD}\) is a median, it divides \(\overline{BC}\) into 2 equal parts. Therefore, \(\overline{BD}\) is 1. Now, we can say that \(\triangle BAD\) is equilateral because it is isoceles, and one of the angles is 60. As such, \(\overline{AD}\) is 1.
The centroid divides medians into 2 parts ratio 1:2, so \(\overline{AG}\) is \(\boxed{\frac{2}{3}}\).
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