Use coordinates, distance, and algebra to prove or disprove simple geometric statements.
Use coordinates to prove a segment is an altitude
Problem
For triangle \(A(0,3)\), \(B(-2,0)\), \(C(4,0)\), test whether \(AD\) with \(D(0,0)\) is an altitude. Verify that \(D\) lies on \(BC\), compare the directions of \(AD\) and \(BC\), and state the conclusion.
An altitude check has two independent requirements: the proposed foot must lie on the line containing the side opposite the vertex, and the vertex-to-foot segment must be perpendicular to that line. Verify membership from the coordinates first, then compare directions, handling a horizontal-and-vertical pair directly rather than multiplying an undefined slope.
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