Consider a triangle \( \triangle ABC \) with vertices \( A(2, 3) \), \( B(8, 3) \), and \( C(5, 9) \). The line \( y = mx + c \) passes through the vertex \( A \) and is perpendicular to the line segment \( BC \). Find the area of \( \triangle ABC \). Let's think step by step and output the final answer within \boxed{}.
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