In the rectangular parallelepiped ABCDA1B1C1D1, the lengths of the ribs are known

| July 23, 2021 | education

In the rectangular parallelepiped ABCDA1B1C1D1, the lengths of the ribs are known: AB = 12, AD = 16, CC1 = 9. find the angle between planes BDD1 and AB1D1.

Section BDD1 is a diagonal section BB1D1D, and section AB1D1 is a triangle AB1D1.

The sections have a common side B1D1.

From point A, draw a perpendicular to the diagonal B1D1. The projection of the segment AH1 onto the base is the segment AH, which is the height of the right-angled triangle ABD.

The angle AH1H is our desired angle.

Determine the area of ​​the triangle ABD.

Savd = AB * AD / 2 = 12 * 16/2 = 96 cm2.

Let us define, by the Pythagorean theorem, the diagonal ВD.

BD ^ 2 = AD ^ 2 + AB ^ 2 = 256 + 144 = 400.

ВD = 20 cm.

Then Savd = 96 = AH * BD / 2.

AH = 96 * 2/20 = 9.6 cm.

The length of the segment НН1 = CC1 = 9 cm.

Then tgAH1H = AH / HH1 = 9.6 / 9 = 1.07.

Then the angle AH1H = arctg1.07 ≈ 47.

Answer: The angle between the planes is 47.



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