In a right-angled triangle KRE, angle P = 90, angle K = 60. Point M was marked on the leg PE

In a right-angled triangle KRE, angle P = 90, angle K = 60. Point M was marked on the leg PE so that the angle KMP = 60. Find PM if EM = 16 cm.

Determine the value of the angle REK of the triangle KPE. Angle PЕК = (90 – 60) = 30.

The KPM triangle is rectangular, then the angle PKM = (90 – KMP) = (90 – 60) = 30.

Since the angle is K = 60, and the angle is PKM = 30, then KM is the bisector of the angle PKE, and then the triangle KME is isosceles, since the angles at the base of KE are equal, which means KM = EM = 16 cm.

In a right-angled triangle KPM, the desired leg PM lies against an angle of 30, then PM = KM / 2 = 16/2 = 8 cm.

Answer: The length of the PM segment is 8 cm.