From one point to the plane, a perpendicular equal to 12 cm and two oblique lines are drawn.

From one point to the plane, a perpendicular equal to 12 cm and two oblique lines are drawn. Calculate the lengths of their projections onto a given plane, if the angle between the plane and each inclined one is respectively 45º and 60º.

We have two right-angled triangles, in which the oblique are the hypotenuses, their projections onto the plane and the perpendicular to the plane are the legs. The inclined projection is the leg adjacent to the angle between this inclined and the plane, the perpendicular is the opposite leg to this angle. The ratio of the opposite leg to the adjacent leg is the tangent of the angle, which means that the adjacent legs can be found as the ratio of the length of the perpendicular to the tangent. Therefore, one of the projections is 12 / tg45 = 12/1 = 12 cm, the other is 12 / tg60 = 12 / √3 = 4√3≈6.93 cm.