From points M and N, the edges of the dihedral angle in its different faces were erected perpendiculars MK

| July 13, 2021 | education

From points M and N, the edges of the dihedral angle in its different faces were erected perpendiculars MK and NL. Determine the size of the dihedral angle, considering that MN = 48 cm, MK = 16 cm, NL = 10 cm and the distance between points K and L is 50 cm.

Given: MN = 48 cm, MK = 16 cm, NL = 10 cm and KL = 50 cm. It is required to determine the value of the dihedral angle.
Let us draw one more perpendicular РМ to the edge MN so that МР || NL and MP = NL = 10 cm.
It is clear that the magnitude of the dihedral angle is equal to the magnitude of the linear angle PMK. In addition, PL = MN = 48 cm.
Let’s connect the points P and K with a segment RK. Since, KM ┴ MN and МР || NL, the angle KPL is straight. Therefore, ΔKPL is a right-angled triangle with a hypotenuse KL and legs KP, PL.
According to the Pythagorean theorem, KL ^ 2 = KP ^ 2 + PL ^ 2, whence KP ^ 2 = KL ^ 2 – PL ^ 2 = (50 cm) ^ 2 – (48 cm) ^ 2 = 196 cm2. Thus, we determined that KR = 14 cm.
We apply the cosine theorem to the triangle PMK: KP ^ 2 = MK ^ 2 + MP ^ 2 – 2 * MK * MP * cos∠PMK. We have: (14 cm) ^ 2 = (16 cm) ^ 2 + (10 cm) ^ 2 – 2 * (16 cm) * (10 cm) * cos∠PMK, whence cos∠PMK = (256 + 100 – 196 ) / 320 = 160/320 = ½.
If cos∠PMK = ½, then according to the table of basic values ​​of sine, cosine, tangent and cotangent ∠PMK = 60 °. This means that the value of this dihedral angle is 60 °.
Answer: 60 °.



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