From one point, a perpendicular and two oblique lines are drawn to this straight line.

From one point, a perpendicular and two oblique lines are drawn to this straight line. Determine the length of the perpendicular if the slopes are 41 and 50 cm. And from the projection onto this line, refer to 3:10.

Formed 2 right-angled triangles, where the oblique lines are hypotenuses. Let us express the legs lying on a straight line as 3x and 10x. The length of the perpendicular (or leg in this rectangle) is expressed as y. We compose 2 equations according to the Pythagorean theorem for each triangle.
41 ^ 2 = 9x ^ 2 + y ^ 2 and 50 ^ 2 = 100x ^ 2 + y ^ 2, from here we express y
y ^ 2 = 1681-9x ^ 2 and y ^ 2 = 2500-100x ^ 2, equate the equations
1681-9x ^ 2 = 2500-100x ^ 2
-819 = -91x ^ 2
9 = x ^ 2, x = 3, substitute into the equation
y ^ 2 = 1681-9 * 3 ^ 2
y ^ 2 = 1600
y = 40 – perpendicular length