Through the vertex C of the regular triangle ABC, in which AC = 16 cm, the perpendicular PC is drawn to the plane of the triangle. Find the angle between the planes ABC and APB, if PB = 20 cm.
Since the base is a regular triangle, AB = AC = BC = 16 cm.
The height CD of a regular triangle is determined by the formula CD = a * √3 / 2, where a is the length of the side of the triangle. СD = 16 * √3 / 2 = 8 * √2.
Consider a right-angled triangle PCB, in which the hypotenuse PC = 20 cm, and the leg CB = 16 cm.Then, according to the Pythagorean theorem, PC ^ 2 = PB ^ 2 – CB ^ 2 = 20 ^ 2 – 16 ^ 2 = 400 – 256 = 144.
RS = 12 cm.
Determine the value of the angle РДС.
tgРDC = РС / СD = 12/8 * √2 = 3/2 * √2 = 1.061.
Angle РDC = arctg1,061 ≈ 48.
Answer: Angle РDC = arctg1,061 ≈ 48.
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