The diagonal of a regular quadrangular prism is 9cm, and the side edge is 7cm. Find the rib at the base.

Consider a rectangular triangle CBC1, in which the hypotenuse BC1 = 9 cm, and the leg CC1 = 7 cm, then, according to the Pythagorean theorem, the leg BC ^ 2 = BC1 ^ 2 – CC1 ^ 2 = 9 ^ 2 – 7 ^ 2 = 81 – 49 = 32.

BC = √32 = 4 * √2 cm.

Since, by condition, the prism is correct, there is a square at its base. In a square, the diagonal is equal to the product of the side and the square root of two.

BC = AB * √2.

AB = BC / √2 = 4 * √2 / √2 = 4 cm.

Or by the Pythagorean theorem:

BC ^ 2 = AC ^ 2 + AB ^ 2 = 2 * AB ^ 2.

AB ^ 2 = BC ^ 2/2 = (4 * √2) ^ 2/2 = 16.

AB = 4 cm.

Answer: The side of the base is 4 cm.