how would you draw a counterexample to this?

Melanie said: Every angle bisector in a triangle bisects the opposite side perpendicularly.
A 'counterexample' would show an angle bisector in a triangle that DOESN'T bisect the opposite side perpendicularly.
See my attached drawing of a counterexample.
Both of the triangles that Melanie examined have equal sides on both sides of the angle bisector. That's the only way that the angle bisector can bisect the opposite side perpendicularly. Melanie didn't examine enough different triangles.

Answered by AL2006 | 2024-06-10

A counterexample to the claim is an isosceles triangle, where the angle bisector does not bisect the opposite side perpendicularly. For example, in triangle ABC with vertices at A(0, 0), B(4, 0), and C(2, 3), the angle bisector does not form a right angle with side BC. This illustrates that the claim is not universally valid for all triangles.
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Answered by AL2006 | 2025-04-09