# The height BM, drawn from the top of the rhombus angle ABCD, forms an angle of 30 degrees with the side AB

The height BM, drawn from the top of the rhombus angle ABCD, forms an angle of 30 degrees with the side AB, AM = 4cm. Find the length of the diagonal BD of the rhombus if point M lies on the side AD

Let’s write all the angles obtained inside the rhombus. <BAD = 60, <ABM = 30, <BCD = <BAD = 60. <DBC = <BDC = (180 – 60) / 2 = 120/2 = 60.
Such calculations were made because the triangles BCD and ABD are equilateral, since AB = AD = BC = CD, as the sides of the rhombus and the acute angles between the sides are equal to 60 by condition.

First, we find the side of the rhombus: AB = DC = CD = AD = AM * 2 = 4 * 2 = 8 (cm), like the hypotenuse in a triangle with a leg against an angle of 30 degrees.
But from the triangle BCD it is clear that all its sides are equal, like the sides in an equilateral triangle. Hence, BD = BC = CD = 8 (cm).

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