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.
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