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

The height of the BM drawn from the apex of the rhombus ABCD forms an angle of 30 degrees with the side AB. AM 4 cm. Find the length of the AD diagonal.

The height of the BM forms a right-angled triangle ABM, in which the angle ABM, by condition, is 30. Then the leg AM lies opposite the angle 30, and therefore, its length is equal to half the length of the hypotenuse AB.
AM = AB / 2, then AB = 2 * AM = 2 * 4 = 8 cm.
In a rhombus, the lengths of all edges are equal, then AD = AB = 8 cm.
We define the BD diagonal by the cosine theorem.
In the AED triangle, the angle BAD = 90 – 30 = 60.
Then BD^2 = AB^2 + AD^2 – 2 * AB * AD * Cos60 = 64 + 64 – 2 * 8 * 8 * (1/2) = 64.
BD = 8 cm.
Answer: The length of the diagonal BD is 8 cm, the AD is 8 cm.