In the regular quadrangular pyramid SABCD, point O is the center of the base, S is the vertex, SO = 12, SB = 15. Find the length of the line segment AC.
Let us construct the AC and ВD diagonals at the base of the pyramid.
In a right-angled triangle SOB, according to the Pythagorean theorem, we determine the length of the leg OB.
OB ^ 2 = SB ^ 2 – SO ^ 2 = 15 ^ 2 – 12 ^ 2 = 225 – 144 = 81.
OB = 9 cm.
Since the base of a regular quadrangular pyramid is the square ABCD, point O divides the diagonals AC and BD in half, then BD = 2 * OB = 2 * 9 = 18 cm.
Since BD = AC, then AC = 18 cm.
Answer: The length of the AC segment is 18 cm.
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