From this point to the plane, a perpendicular with a length of 1 dm is drawn. and two equal oblique.

From this point to the plane, a perpendicular with a length of 1 dm is drawn. and two equal oblique. find the lengths of the inclined ones if the angle between them is 60 degrees, and their projections are mutually perpendicular.

Since, by condition, the angle BAC = 60, and AB = AC, the triangle ABC is equilateral.

Let AB = AC = BC = X cm.

The triangle AOB is rectangular, then, according to the Pythagorean theorem, OB ^ 2 = AB ^ 2 – AO ^ 2 = X ^ 2 – 100.

Since oblique AB = AB, then their projections OB = OS, and then the triangle ВOС is rectangular and isosceles.

Then, by the Pythagorean theorem, BC ^ 2 = OB ^ 2 + OC ^ 2.

X ^ 2 = X ^ 2 – 100 + X ^ 2 – 100.

X ^ 2 = 200.

X = BC = AB = AC = 10 * √2 cm.

Answer: The lengths of the slopes are 10 * √2 cm.