In triangle ABC, the bisector BH is perpendicular to side AC. Find the length of the line segment AH if HC = 5cm.

1. BH is perpendicular to AC, so BH is the height of triangle ABC (by definition of height)

2. In triangle ABC: BH is the bisector and height of triangle ABC, which means triangle ABC is isosceles (based on an isosceles triangle)

3. Since ABC is an isosceles triangle, then BH is a median (by the property of an isosceles triangle), which means

AH = HC = 5 cm

Answer: AH = 5 cm.