Find the lengths of the sides in the parallelogram if the diagonals 10 and 32 cm long intersect at an angle of 120 °.

| January 15, 2021 | education

In order to solve this problem, you need to find the lengths of the sides in the parallelogram:
1) Consider a triangle AMB, AM – half of the AC diagonal, MB – half of the BD diagonal, ∟AMB – an angle of 120 °.
2) Find MВ and AM: MВ = 32/2 = 16 cm; AM = 10/2 = 5 cm.
3) Now, by the cosine theorem, we find the larger side of the parallelogram: AB ^ 2 = AM ^ 2 + MB ^ 2 – 2 * AM * MB * cos120 °; AB ^ 2 = AM ^ 2 + MB ^ 2 + AM * MB = 25 + 256 + 16 * 5 = 361; AB = 19 cm.
4) The second side can be found by this property: the sum of the squares of the diagonals of a parallelogram is equal to the sum of the squares of its sides.
5) So, the second side of the BC is equal to: 2BC ^ 2 + 2AB ^ 2 = AC ^ 2 + BD ^ 2; 2BC ^ 2 = AC ^ 2 + BD ^ 2 – 2AB ^ 2 = 1024 + 100 – 722 = 402; BC = 14.2 cm.
Therefore, our answer is: 14.2 cm and 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.