The perimeter of the parallelogram is 144cm, the bisector of an acute angle divides its larger side into segments

| February 11, 2021 | education

The perimeter of the parallelogram is 144cm, the bisector of an acute angle divides its larger side into segments, the lengths of which are 3: 6, counting from the apex of the obtuse angle. Find the smaller side of the parallelogram.

Let AВСD be a parallelogram, AE is a bisector (A is an acute angle), E belongs to BC. BE: CE = 3: 6. Perimeter (AВСD) = 144 cm. Find AB.

The angle BEA is equal to the angle DAE (internal criss-crossing angles with parallel AD and BC and secant AE).

The angle DAE is equal to the angle BAE (AE is the bisector of angle A).

This means that the angle BAE is equal to the angle BEA, and therefore the triangle ABE is isosceles (in an isosceles triangle, the angles at the base are equal).

Hence, BE = AB.

Let’s denote the ratio coefficient as x. Then BE = 3x, CE = 6x, BC = 9x.

AB = 3x (since it is equal to BE).

Let us express the perimeter of the parallelogram: P = 3x + 9x + 3x + 9x = 24x.

Since the perimeter is 144 cm, then 24x = 144; x = 144/24 = 6.

Hence AB = 3x = 3 * 6 = 18 cm.

Answer: The smaller side of the parallelogram is 18 cm.



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