The apothem of a regular triangular truncated pyramid is 10 cm. The height of the upper base is 6 cm, the height of the lower base is 24 cm. Find the height of the truncated pyramid.
Since the pyramid is regular, equilateral triangles lie at the bases of the truncated pyramid. The heights of equilateral triangles are also their medians, which are divided in the ratio of 2/1 at the point of intersection.
Then О1Н1 = 6/3 = 2 cm.
OH = 24/3 = 8 cm.
Quadrangle OO1H1H is a rectangular trapezoid in which we draw the height KH1. Then KH = OH – OK = OH – O1H1 = 8 – 2 = 6 cm.
In a right-angled triangle HH1K, according to the Pythagorean theorem, KH1 ^ 2 = HH1 ^ 2 – KH ^ 2 = 100 – 36 = 64.
KH1 = 8 cm, then OO1 = KH1 = 8 cm.
Answer: The height of the truncated pyramid is 8 cm.
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