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

The height BM, drawn from the vertex 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 the point M lies on the side of AD.

Since BM is the height of a rhombus, the ABM triangle is rectangular, in which the ABM angle, by condition, is 30.

Then the angle BAM = (90 – 30) = 60.

Since all the faces of a rhombus are equal, then AB = AD, and then the triangle ABD is isosceles, and since the angle BAD = 60, then the triangle ABD is equilateral.

Then the height of the BM is also the median of the triangle ABD, then AM = DM = 4 cm, AD = BD = 8 cm.

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