Point M is marked in the parallelogram ABCD – the middle of the BC side. The segments BD

Point M is marked in the parallelogram ABCD – the middle of the BC side. The segments BD and AM intersect at point K. Find BK if BD = 12.

Consider a parallelogram ABCD.

By the statement of the problem, M is the midpoint of side BC and BD = 12.

Now consider two triangles: BMK and AKD.

Since AD and BC are parallel, the angle KAM = BMK and KBM = KDA.

Therefore, triangles BMK and AKD have all angles equal and the triangles are similar. Therefore, we have:

BK / KD = BM / AD.

Since M is the middle of the side BC, then BM / AD = 1/2, which means:

BK / KD = 1/2, KD = 2 * BK.

Then:

BK + KD = 12, BK + 2 * BK = 12, 3 * BK = 12, BK = 4.

Answer: BK = 4.