From a point at a distance of 12 cm from the plane, two oblique ones are drawn, forming angles of 45 degrees

From a point at a distance of 12 cm from the plane, two oblique ones are drawn, forming angles of 45 degrees and 60 degrees with the plane. Find the distance between the inclined bases if the angle between their projections is straight.

Since AD is perpendicular to the plane, the triangles ABD and ACD are rectangular.

Angle ABD = 45, then right-angled triangle ABD is isosceles, AB = AD = 12 cm.

In a right-angled triangle ACD tg60 = AD / AC.

AC = AD / tg60 = 12 / √3 = 4 * √3 cm.

AC is perpendicular to AB by condition, then triangle ABC is rectangular.

BC ^ 2 = AB ^ 2 + AC ^ 2 = 144 + 48 = 192.

BC = 8 * √3 cm.

Answer: The distance between the bases of the slopes is 8 * √3 cm.