In the parallelogram ABCD, the bisectrices of angles A and D have been cut through the side BC

| March 19, 2021 | education

In the parallelogram ABCD, the bisectrices of angles A and D have been cut through the side BC into three equal segments BF FE. Find the perimeter of the parallelogram if side BC is 24.

Since AE is the bisector of angle A, the angle ABE = EAD. The angle BEA = EAD as cross-lying angles at the intersection of parallel straight lines AD and BC of the secant AE, then the angle BAE = BEA, and, accordingly, the triangle ABE is isosceles, AB = BE.

Since, by condition, BF = FE = CE, and AC = 24 cm, then BF = FE = CE = 24/3 = 8 cm.Then AB = BF + FE = 8 + 8 = 16 cm.

AB = CD = 16 cm, BC = AD = 24 cm.

Determine the perimeter of the parallelogram.

P = 2 * AB + 2 * AD = 32 + 48 = 80 cm.

Since the condition does not specify whether the bisectors intersect or not, we will consider the second possible case, if the bisectors of the angles do not intersect, then the sides AB and BC will be 8 cm, and the perimeter P = 2 * 8 + 2 * 24 = 16 + 48 = 64 cm.

Answer: The perimeter is 80 cm (64 cm).



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