The obtuse bisector of a parallelogram divides the opposite side in a ratio of 1: 3, counting from the top of the acute angle.

The obtuse bisector of a parallelogram divides the opposite side in a ratio of 1: 3, counting from the top of the acute angle. Find the large side of the parallelogram if its perimeter is 10

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 = СD = X cm.
The perimeter of the parallelogram is: Ravsd = (AB + AD + BC + СD) = (X + 4 * X + 4 * X + X) = 10.
10 * X = 10.
X = 10/10 = 1.
Then AB = СD = 1 cm.
Answer: The side is 1 cm.