In rectangular parallelepiped ABCDA1B1C1D1, the side face of DD1C1C is a square, DC = 3, BD1 = √22. Find BC.

Since, according to the condition, the side face of DD1C1C is square, and DС = 3 cm, then DD1 = DС = 3 cm.

At the base of the parallelepiped lies a rectangle in which we draw the diagonal BD.

Then the triangle BDD1 is rectangular, in which, according to the Pythagorean theorem, we determine the length of the leg BD.

BD ^ 2 = BD1 ^ 2 – DD1 ^ 2 = (√22) ^ 2 – 3 ^ 2 = 22 – 9 = 13.

ВD = √13 cm.

The BCD triangle is also rectangular, then BC ^ 2 = BD ^ 2 – CD ^ 2 = 13 – 9 = 4.

BC = 4 cm.

Answer: The length of the BC rib is 4 cm.