Find the side of the base of a right triangle of a pyramid if its height is 5 cm

Find the side of the base of a right triangle of a pyramid if its height is 5 cm and its lateral edge is inclined to the base plane at an angle of 45 degrees.

Consider a right-angled triangle AOН, in which the angle AНO is straight, and the angle НAO = 45, then the angle AOH = 180 – 90 – 45 = 45, which means that the triangle AOH is equilateral and OH = AH = 5 cm.

Since there is an equilateral triangle at the base, then AH / KH = 2/1, then KH = AH / 2 = 5/2 cm.

The segment KН is the radius of a circle inscribed in an equilateral triangle, which is determined by the formula: r = a * √3 / 6, where a is the side of an equilateral triangle, then

a = 6 * r / √3 = 6 * (5/2) / √3 = 15 / √3 = 5 * √3 cm.

Answer: the side of the base is 5 * √3 cm.