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.
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