In $\triangle ABC$, $\angle BAC = 2x$ and $\triangle ABC$ is isosceles with $AB = AC = 10$ units. The height from $B$ to side $AC$ is 8 units. Using the properties of isosceles triangles and trigonometric identities, find the area of $\triangle ABC$. Let's think step by step and output the final answer within \boxed{}.
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