In triangle ABC, angle C = 90 degrees, angle A = 30 degrees, segment BM is the bisector of the triangle.

In triangle ABC, angle C = 90 degrees, angle A = 30 degrees, segment BM is the bisector of the triangle. Find the length of the AC leg if BM = 6cm.

Determine the value of the angle ABC. Angle ABC = (90 – BAC) = (90 – 30) = 60.

Since, by condition, BM is the bisector of the angle, then the angle ABM = CBM = ABC / 2 = 60/2 = 30.

Then in the triangle ABM the angles at the base of AB are equal, and therefore, the triangle ABM is isosceles, AM = BM = 6 cm.

The BCM triangle is rectangular, in which the CBM angle = 60, then the length of the CM leg, lying opposite it, is equal to: CM = BM / 2 = 6/2 = 3 cm.

Leg AC = AM + CM = 6 + 3 = 9 cm.

Answer: The length of the AC leg is 9 cm.