The parallelogram ABCD marks the midpoint of side BC. BD and AM intersect at point K. Find BK if BD = 12.

Triangles BKM and AKD are similar in two angles, since the angle BKM = AKM as vertical angles at the intersection of straight lines AM and BD, the angle BMK = KAD as criss-crossing angles at the intersection of parallel lines BC and BD secant AM.

Since, on the condition, BM = CM = BC / 2, the coefficient of similarity of triangles is equal to:

K = BM / AD = 1/2.

Then BK / DK = 1/2.

BK + DK = BD.

DK = BK * 2.

BK + BK * 2 = BD.

3 * BK = BD.

BK = BD / 3 = 12/3 = 4 cm.

Answer: The length of the segment BK is 4 cm.