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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