In a regular triangular pyramid, the lateral edge is 5 comma, the tangent of the angle between the lateral

| June 22, 2021 | education

In a regular triangular pyramid, the lateral edge is 5 comma, the tangent of the angle between the lateral face and the plane of the base is 0.25√11 find the side of the base of the pyramid.

Since an equilateral triangle lies at the base of the pyramid, then AO / OH = 2/1.

Let the length of the segment OH = X cm, then AO = 2 * X cm.

In a right-angled triangle AOD, according to the Pythagorean theorem, DO ^ 2 = AD ^ 2 – AO ^ 2 = 25 – 4 * X2. (one).

In a right-angled triangle DOH, tgH = DО / ОН.

DO = OH * tgH = X * √11 / 4.

Then DO ^ 2 = X ^ 2 * 11 / 16. (2).

Equate Equations 1 and 2.

25 – 4 * X ^ 2 = 11 * X ^ 2/16.

75 * X ^ 2/16 = 25.

X ^ 2 = 25 * 16/75

X = 20/5 * √3 = 4 / √3 cm

AO = 8 / √3.

AH = 4 / √3 + 8 / √3 = 12 / √3 = 4 * √3 cm.

AH = BC * √3 / 2.

BC = 2 * AH / √3 = 2 * 4 * √3 / √3 = 8 cm.

Answer: The side of the base of the pyramid is 8 cm.



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