The bisector of the angle A of the parallelogram ABCD divides the side CD in the ratio 1: 3 to the vertex C

The bisector of the angle A of the parallelogram ABCD divides the side CD in the ratio 1: 3 to the vertex C, find the sides if the perimeter is 84cm.

Let’s designate the point of intersection of the bisector with the side CD – point M. ADM triangle is isosceles, since angles <MAD = <MDA = <BAM, since AB is parallel to CD, and AM is bisector AD = DM, CM = 3 * MD, CD = AB = MD + CM = 4 * MD = 4 * AD.
Considering that the perimeter of the parallelogram is ABCD = AB + BC + CD + AD = 4 * AD + AD + 4 * AD + AD = 10 * AD = 84 As a result, we find AD = 84/10 = 8.4 (cm). AB = CD = 4 * AD = 4 * 8.4 = 33.6 (cm).