Line AB intersects the alpha plane at an angle of 30 degrees. AA1 is the perpendicular, and BA1 is the projection

Line AB intersects the alpha plane at an angle of 30 degrees. AA1 is the perpendicular, and BA1 is the projection of AB on the alpha plane. Find: 1) BA1 and AA1, if AB = 24 cm, 2) The length of the projection BA1 of oblique AB, if AA1 = 8 cm , 3) The length of the oblique AB and the length of the perpendicular AA1, if BA1 = 15 cm

The segment AA1 is perpendicular to the plane, and therefore perpendicular to the projection BA1, then the triangle ABA1 is rectangular.

1. Cathet AA1 lies opposite angle 300, then its length is equal to half the length of the hypotenuse AB. AA1 = AB / 2 = 24/2 = 12 cm.

CosABA1 = BA1 / AB.

BA1 = AB * Cos30 = 24 * √3 / 2 = 12 * √3 cm.

Answer: The AA1 segment is 12 cm, the BA1 segment is 12 * √2 cm.

2.In a right-angled triangle ABA1 tg30 = AA1 / BA1.

BA1 = AA1 / tg30 = 8 / (√3 / 3) = 24 / √3 = 8 * √3 cm.

Answer: The projection length is 8 * √3 cm.

In a right-angled triangle ABA1 tg30 = AA1 / BA1.
АA1 = ВA1 * tg30 = 15 * (√3 / 3) = 5 * √3 cm.

AB2 = AA12 + BA12 = 75 + 225 = 300.

AB = 10 * √3 cm.

Answer: The length of the oblique AB is 10 * √3 cm, the length of AA1 is 5 * √3 cm.

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