# Given a quadrangular pyramid at the base of which lies a square with a diagonal of 8√2

**Given a quadrangular pyramid at the base of which lies a square with a diagonal of 8√2. Find the apothem of the pyramid if its height is 3cm**

Since the pyramid is correct, there is a square at its base, and the top of the pyramid is projected to point O, the point of intersection of the diagonals AC and BD.

By the Pythagorean theorem, AC ^ 2 = AD ^ 2 + CD ^ 2 = 2 * AD ^ 2.

AD ^ 2 = AC ^ 2/2 = 64 * 2/2 = 64.

AD = 8 cm.

The lateral faces of the pyramid are isosceles triangles, then the apothem PH is the height and median of the PCD triangle.

Point O is the middle of AC, point H is the middle of CD, then OH is the middle line of the triangle ACD.

OH = AD / 2 = 8/2 = 4 cm.

By the Pythagorean theorem, in a right-angled triangle POH, we define the length of the hypotenuse PН.

PH ^ 2 = PO ^ 2 + OH ^ 2 = 9 + 16 = 25.

PH = 5 cm.

Answer: The length of the apothem is 5 cm.