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.
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