The height of a regular quadrangular pyramid is 7cm, and the side of the base is 8cm. Find the length of the side rib.

Let’s draw the diagonals AC and BD at the base of the pyramid.

Consider a triangle ABD, in which AB = AD, since there is a square at the base of the pyramid.

Then, by the Pythagorean theorem, BD ^ 2 = AB ^ 2 + AD ^ 2 = 8 ^ 2 + 8 ^ 2 = 64 + 64 = 128.

ВD = √128 = 8 * √2 cm.

AC = BD = 8 * √2 cm.

Then OA = AC / 2 = 8 * √2 / 2 = 4 * √2 cm.

Consider a right-angled triangle AMO and, by the Pythagorean theorem, determine the length of the hypotenuse AM.

AM ^ 2 = AO ^ 2 + MO ^ 2 = (4 * √2) ^ 2 + 7 ^ 2 = 32 + 49 = 81.

AM = √81 = 9 cm.

Answer: The length of the side edge of the pyramid is 9 cm.