In the parallelogram ABCD: AB = 4cm, AD = 7cm, angle A = 60 degrees. The diagonals of the parallelogram

| July 23, 2021 | education

In the parallelogram ABCD: AB = 4cm, AD = 7cm, angle A = 60 degrees. The diagonals of the parallelogram intersect at point O, the segment OM is perpendicular to the plane ABC and OM = 5cm. Determine the lengths of the MC and MD segments.

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

BD ^ 2 = AB ^ 2 + AD ^ 2 – 2 * AB * AD * Cos60 = 16 + 49 – 2 * 4 * 7 * (1/2) = √37 cm.

The diagonals of the parallelogram at point O are divided in half, then OD = BD / 2 = √37 / 2 cm.

In a right-angled triangle MOD, according to the Pythagorean theorem, MD ^ 2 = OM ^ 2 + OD ^ 2 = 25 + 37/4 = (100 + 37) / 4.

МD = √137 / 2 cm.

In the triangle ABC, by the cosine theorem, we define the length of the diagonal AC.

AC ^ 2 = AB ^ 2 + BC ^ 2 – 2 * AB * BC * Cos120 = 16 + 49 – 2 * 4 * 7 * (-1/2) = √93 cm.

OS = AC / 2 = √93 / 2 cm.

In a right-angled triangle MOS, according to the Pythagorean theorem, MC ^ 2 = OM ^ 2 + OC ^ 2 = 25 + 93/4 = (100 + 93) / 4.

MC = √193 / 2cm.

Answer: The lengths of the segments are equal to √137 / 2 cm and √193 / 2 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.