The diagonals of the rhombus ABCD meet at point O. Calculate the degree measures of the acute angles

The diagonals of the rhombus ABCD meet at point O. Calculate the degree measures of the acute angles of the triangle AOB if BD = 10 cm and the area of the rhombus is 50√3 cm2.

Let us determine the length of the second diagonal in terms of the area of the rhombus.

Savsd = АС * ВD / 2.

АС = 2 * Savsd / ВD = 2 * 50 * √3 / 10 = 10 * √3 cm.

The diagonals of the rhombus, at the point of their intersection, are halved and intersect at right angles.

Then ОВ = ОD = ВD / 2 = 10/2 = 5 cm, AO = СО = АС / 2 = 10 * √3 / 2 = 5 * √3 cm, and triangle AOB is rectangular.

tgOAB = OB / AO = 5/5 * √3 = √3 / 3.

Angle OAB = arctan (√3 / 3) = 30, then angle ABO = 90 – 30 = 60.

Answer: The angles of triangle AOB are 30 and 60.