Angle A in rhombus ABCD is 30. Perpendiculars BM and BK are drawn from vertex B to sides AD and CD, respectively. What is the sum of the lengths of the segments MD and CK.
Let us prove that triangle ABM is equal to triangle BCK.
Triangles ABM and BCK are rectangular, since the segments BM and BK, by condition, are the height of the rhombus.
In a rhombus, the opposite sides are equal, then the angle BСD = BAD = 30.
In a rhombus, all sides are equal, AB = BC = CD = AD.
Then triangle ABM is equal to triangle BCK in hypotenuse and acute angle.
Then AM = CK.
But since AD = CD, then MD = KD.
Then MD + CK = AD + AM = AD.
Answer: The sum of the segments MD and CK is equal to the length of the side of the rhombus.
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