In parallelogram ABCD, the bisector of angle B intersects side AD at point M

In parallelogram ABCD, the bisector of angle B intersects side AD at point M so that AM is 4 times MD. Find the lengths of the sides of the parallelogram if its perimeter is 36 cm

1. The bisector CM of the parallelogram ABCD cuts off the triangle CDM from it, which is isosceles. Therefore, DМ = СD.

2. AM = 4DM = 4CD.

3. AD = AM + DM. We replace in this expression AM with 4CD, DM with C:

AD = 4CD + CD = 5CD.

4. Considering that the opposite sides of the parallelogram are equal, its perimeter is calculated by the formula:

2AD + 2СD = 36 cm.

AD + CD = 18 cm.

5СD + СD = 18 cm.

6СD = 18 cm.

CD = 3 cm.

AD = 3 x 5 = 15 cm.

Answer: AD = BC = 15 cm, AB = CD = 3 cm.