In the parallelogram ABCD, the bisector of angle A intersects the side BC at point E. The segment BE

In the parallelogram ABCD, the bisector of angle A intersects the side BC at point E. The segment BE is 3 times larger than the segment EC. Find the perimeter of the parallelogram if BC = 12 cm.

P ABCD = AB + BC + CD + AD

BC = AD = 12; AB = CD – the opposite sides of the parallelogram are equal.

Find the lengths of the sides AB and CD:
The angles BAE and EAD are equal, because AE – bisector of angle A; the angles BEA and EAD are equal as the internal criss-crossing angles with parallel lines BC and AD and secant AE. This means that the angles BEA and BAE will be equal and therefore the triangle ABE will be isosceles. In an isosceles triangle, the sides are equal, which means AB = BE.
Let CE early x cm, then BE – (3x) cm.Their sum is (x + 3x) cm or 12 cm.

x + 3x = 12;

4x = 12;

x = 12: 4;

x = 3 (cm) – CE;

3x = 3 * 3 = 9 (cm) – BE.

AB = CD = 9 cm.

P ABCD = 9 + 12 + 9 + 12 = 42 (cm).

Answer. 42 cm.