The height BM, drawn from the vertex of the corner of the rhombus ABCD, forms an angle of 30 °

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

Triangle ABM is rectangular, since BM is the height of a rhombus, then in a right-angled triangle ABM Sin30 = AM / AB.

AB = AM / Sin30 = 4 / (1/2) = 8 cm.

In a rhombus, all sides are equal, then AB = AD = 8 cm.

To find the length of the diagonal BD, we prove that triangle ABD is equilateral.

Angle BAM = BAD = 180 – 90 – 30 = 60.

Since in the triangle ABD, AB = BD, and the angle BAD = 60, then the triangle is equilateral BD = AB = 8 cm.

The length BD can also be determined by the cosine theorem.

BD ^ 2 = AB ^ 2 + AD ^ 2 – 2 * AB * AD * Cos60 = 64 + 64 – 2 * 64/2 = 64.

ВD = 8 cm.

Answer: The length of the diagonal BD is 8 cm.