The bisector and height are drawn from vertex A of triangle ABC. Determine which of these segments

The bisector and height are drawn from vertex A of triangle ABC. Determine which of these segments can be larger than the AC side.

The height АН is the leg of the right-angled triangle АСН, and therefore cannot be more than the side АС of the triangle ABC.

Consider a triangle AFM, where AM is the bisector of angle A.

The larger side is opposite the larger corner of the triangle.

Then AM> AC if the angle is AFM> AMC.

Angle AMB = (180 – B – A / 2), then angle AMC = (180 – (180 – B – A / 2)).

If the angle is AFM> (B + A / 2), then AM> AC.

Answer: The bisector AM can be larger than the AC side.