In parallelogram ABCD, angle A = 60 °, diagonal BD is perpendicular to side AB. A straight line passing through

univerkov | September 25, 2021 | education

In parallelogram ABCD, angle A = 60 °, diagonal BD is perpendicular to side AB. A straight line passing through the middle of the segment – point M – parallel to AD, intersects side AB at point K, MK = 4cm – Find S of parallelogram ABCD – Find S of triangle AMD.

Since the segment KM is parallel to the base of AD, as well as point M is the middle of the diagonal BD, the segment KM is the middle line of the triangle ABD, and therefore AD = 2 * KM = 2 * 4 = 8 cm.

In a right-angled triangle ABD, the angle ADB = 180 – 90 – 60 = 30.

Leg AB lies opposite angle 30, and therefore is equal to half the length of AD. AB = AD / 2 = 8/2 = 4 cm. Determine the area of the parallelogram.

Savsd = AB * AD * Sin60 = 4 * 8 * √3 / 2 = 16 * √3 cm2.

In a right-angled triangle ABD, we determine the length of the leg BD. SinBAD = BD / AD.

ВD = АD * SinВD = 8 * √3 / 2 = 4 * √3 cm, then the segment МD = ВD / 2 = 4 * √3 / 2 = 2 * √3 cm.

Let’s define the area of the triangle АМD. Samd = (1/2) * AD * MD * Sin30 = (1/2) * 8 * 2 * √3 * 1/2 = 4 * √3 cm2.

Answer: The area of the parallelogram is 16 * √3 cm2, the area of the triangle АМD is 4 * √3 cm2.

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.