In a right-angled triangle from the vertex of an angle equal to 60 degrees, a bisector is drawn

In a right-angled triangle from the vertex of an angle equal to 60 degrees, a bisector is drawn, the length of which is 18 cm. Find the length of the leg, which lies opposite the given angle.

Since the segment AM is the bisector of the angle CAB, the angle CAM = CAB / 2 = 60/2 = 30.
In a right-angled AСМ triangle, the CM leg is located opposite an angle of 30, then CM = AM / 2 = 18/2 = 9 cm.
Angle BAM = CAB / 2 = 60/2 = 30, angle ABM = (90 – BAC) = (90 – 60) = 30.
Then in triangle ABM the angles at the base of AB are equal to 30, then ABM is an isosceles triangle, then BM = AM = 18 cm.
Leg length BC = CM + BM = 9 + 18 = 27 cm.
Answer: The length of the leg is 27 cm.