In a rectangular parallelepiped, the sides of the base are 10 and 17 cm, one of the diagonal of the base is 21 cm

| June 17, 2021 | education

In a rectangular parallelepiped, the sides of the base are 10 and 17 cm, one of the diagonal of the base is 21 cm. The large diagonal of the parallelepiped is 29 cm. Find the side edge of the parallelepiped.

Determine which of the diagonals of the base of the parallelepiped is the smallest.

Let the diagonal AC = 21 cm.

We use the property of a parallelogram, according to which the sum of the squares of the diagonals is equal to the doubled sum of the square of the lengths of its sides.

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

BD ^ 2 = 2 * (AB ^ 2 + BC ^ 2) – AC ^ 2 = 2 * (100 + 289) – 441 = 778 – 441 = 337.

BD = √337 = 18.36 cm.

The AC diagonal is larger than BD, then it is the projection of the larger parallelepiped diagonal onto the base plane CA1 = 29 cm.

In a right-angled triangle ACA1 according to the Pythagorean theorem, AA1 ^ 2 = CA1 ^ 2 – AC ^ 2 = 841 – 441 = 400.

AA1 = 20 cm.

Answer: The height of the parallelepiped is 20 cm.



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