From the point of the plane, a perpendicular and an inclined line are drawn.

From the point of the plane, a perpendicular and an inclined line are drawn. The inclined one draws an angle of 45 degrees with the plane, the projection of the inclined 6 cm. Find the length of the inclined.

Let the point from which the perpendicular and the oblique are drawn is point A. Point B is the point of intersection of the oblique with the plane, and point C is the point of intersection of the perpendicular with the plane. Thus, a right-angled triangle ABC was formed with a hypotenuse AB, legs AC and BC = 6 cm and a right angle C = 90 degrees. Angle B is 45 degrees by convention.
Consider a triangle ABC.
From the theorem on the sum of the angles of a triangle, it is known that the sum of all the angles of any triangle is 180 degrees. Then:
angle A + angle B + angle C = 180 degrees;
angle A + 45 degrees + 90 degrees = 180 degrees;
angle A = 180 degrees – 135 degrees;
angle A = 45 degrees.
In triangle ABC, two angles A and B are equal to each other, we can conclude that ABC is an isosceles triangle. Then AB is the base of an isosceles triangle, and the legs AC and BC are equal to each other, that is, AC = BC = 6 cm.
By the Pythagorean theorem, we find the length AB:
AB = √ (BC ^ 2 + AC ^ 2) = √ (6 ^ 2 + 6 ^ 2) = √ (36 + 36) = 6√2 (cm).
Answer: AB = 6√2 cm.