In triangle \(ABC\) above, \(\overline{AD}\) bisects \(\overline{BC}.\) Given that \(AB = 12,\) \(AD =8,\) and \(\sin \angle BAD = \frac{1}{4},\) what is the area of \(\triangle ABC?\)

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In triangle \(ABC,\) \(\overline{AD}\) bisects \(\angle BAC,\) \(\overline{AE}\) is perpendicular to \(\overline{BC},\) and \(F\) and \(G\) are the midpoints of \(\overline{AB}\) and \(\overline{AC},\) respectively. Given that \(AB = 16,\) \(AC=13,\) \(AH=13,\) and \(AI=8,\) and \(\angle BAC = 90^{\circ},\) which of the following triangles has an area equal to half the area of \(\triangle ABC?\)

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