In a rectangular parallelepiped, the sides of the base are 10 cm and 6 cm.The smaller diagonal of the base is 8 cm

In a rectangular parallelepiped, the sides of the base are 10 cm and 6 cm.The smaller diagonal of the base is 8 cm, and the smaller diagonal of the parallelepiped is inclined to the plane of the base at an angle of 60 degrees? Find its second diagonal.

Consider a right-angled triangle DBB1, in which, according to the condition, the angle BDB1 = 60, BD = 8 cm, then the angle BB1D = 90 – 60 = 30, and the leg BD lies opposite the angle 30, then DB1 = 2 * BD = 2 * 8 = 16 cm.

BB1 ^ 2 = DB1 ^ 2 – BD ^ 2 = 256 – 64 = 192.

BB1 = √192 = 8 * √3 cm.

At the base of a parallelepiped lies a parallelogram, then the sum of the squares of the lengths of its diagonals is equal to the sum of the squares of the lengths of its sides.

AC ^ 2 + BD ^ 2 = 2 * (AB ^ 2 + AD ^ 2).

AC ^ 2 = 2 * (36 + 100) – 64 = 272 – 64 = 208.

AC = √208 cm.

From a right-angled triangle ACC1, AC1 ^ 2 = AC ^ 2 + CC1 ^ 2 = 208 + 192 = 400.

AC1 = 20 cm.

Answer: The length of the second diagonal is 20 cm.




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