In the MNPK parallelogram, the MP and NK diagonals are 20 cm and 8 cm

In the MNPK parallelogram, the MP and NK diagonals are 20 cm and 8 cm, respectively. Find the sides of the parallelogram if the diagonals form an angle of 60 degrees.

Since the MНРK is a parallelogram, its diagonals, at the point of their intersection, are divided in half.

Then MO = PO = MР / 2 = 20/2 = 10 cm, HO = KO = НK / 2 = 8/2 = 4 cm.

In the triangle MOН, according to the cosine theorem, MH ^ 2 = MO ^ 2 + HO ^ 2 – 2 * MO * HO * Cos60 = 100 + 16 – 2 * 10 * 4 * (1/2) = 116 – 40 = 76.

MH = 2 * √19 cm.

In the triangle HOP, by the cosine theorem, HP ^ 2 = PO ^ 2 + HO ^ 2 – 2 * PO * HO * Cos120 = 100 + 16 – 2 * 10 * 4 * (-1/2) = 116 + 40 = 156.

HP = 2 * √39 cm.

Answer: The sides of the parallelogram are 2 * √19 cm and 2 * √39 cm.