In the parallelogram ABCD, the angle BAD = 60 degrees. The bisector AT of the angle BAD intersects the side BC

| March 25, 2021 | education

In the parallelogram ABCD, the angle BAD = 60 degrees. The bisector AT of the angle BAD intersects the side BC at point T. It is known that AD = 15 cm, BT = 10 cm. Calculate the lengths of the diagonals of the parallelogram.

Since AT is the bisector of the angle BAD, the triangle ABT is isosceles, AB = BT = 10 cm.

In triangle ABD, by the cosine theorem, we define the length of the diagonal BD.

BD ^ 2 = AB ^ 2 + AD ^ 2 – 2 * AB * AD * Cos60 = 100 + 225 – 2 * 10 * 15 * (1/2) = 325 – 150 = 175.

ВD = 5 * √7 cm.

The sum of the squares of the lengths of the diagonals of a parallelogram is equal to twice the sum of the squares of the lengths of its sides.

AC ^ 2 + BD ^ 2 = 2 * (AB ^ 2 + AD ^ 2).

AC ^ 2 = 2 * (AB ^ 2 + AD ^ 2) – BD ^ 2 = 2 * (100 + 225) – 175 = 650 – 175 = 475.

AC = 5 * √19 cm.

Answer: The lengths of the diagonals are 5 * √7 cm and 5 * √19 cm.



One of the components of a person's success in our time is receiving modern high-quality education, mastering the knowledge, skills and abilities necessary for life in society. A person today needs to study almost all his life, mastering everything new and new, acquiring the necessary professional qualities.