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

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

Let a rhombus ABCD be given, in which the height BM drawn from the vertex ∠ABC forms ∠ABM = 30 ° with side AB, segment AM = 4 cm, then:

4 ∙ 2 = 8 (cm) – the length of the hypotenuse AB in the right-angled triangle ABM (∠BMA = 90 °), by the property of the leg opposite to ∠ABM = 30 °, then the side of the rhombus AB = 8 cm;

8 – 4 = 4 (cm) the length of the segment МD, since by the property of the relative position of points on the straight line АD = АМ + МD.

ΔАВМ = ΔDВМ pr 1 sign of equality of right-angled triangles (along two legs):

2) BM – common leg;

2) AM = MD = 4cm.

Therefore, the hypotenuses of the triangles will be equal to AB = BD = 8 cm and the length of the diagonal of the rhombus BD = 8 cm.

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