The bisector of an obtuse angle of a parallelogram divides the side in a ratio of 1: 3, find the side if its half-perimeter is 55.

| September 4, 2021 | education

The perimeter of the parallelogram is: P = 55 * 2 = 110 cm.

Let the length of the segment DM = 3 * X cm, then, by condition, the length of the segment AM = X cm. Side length AD = (AM + DM) = (X + 3 * X) = 4 * X cm.

Since BM is the bisector of the angle ABC, the triangle ABM is isosceles, since the angle ABM = AMB. Then AB = AM = X cm.

By the property of the parallelogram BC = AD = 4 * X cm, AB = SD = X cm.

The perimeter of the parallelogram is: Ravsd = (AB + AD + BC + CD) = (X + 4 * X + 4 * X + X) = 110.

10 * X = 110.

X = 110/10 = 11.

Then AB = CD = 11 cm.

Since the condition does not indicate from which of the angles the base divides the bisector, the second option is when DM = X cm, and AM = 3 * X cm.

Then AB = 3 * Xcm, and the perimeter of the parallelogram will be: Ravsd = (3 * X + 4 * X + 3 * X + 4 * X) = 14 * X = 110 cm.

X = 110/14 = 55/7 cm.

Then AB = CD = 3 * 55/7 = 23 (4/7) cm.

Answer: The side is 11 cm or 23 (4/7) cm.



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