A square is inscribed in a right-angled triangle, which has a common right angle with the triangle.

A square is inscribed in a right-angled triangle, which has a common right angle with the triangle. One of the vertices of the square lies on the middle hypotenuse. Find the perimeter of the square if the length of the hypotenuse is 24√2.

Since point M is the middle of the hypotenuse AB, then the Segments KM and HM are the middle lines of the triangle ABC, then AC = 2 * HM, BC = * KM.

Since KM and HM are equal as the sides of the square, then AC = BC, then the triangle ABC is rectangular and equilateral.

By the Pythagorean theorem, AB ^ 2 = AC ^ 2 + BC ^ 2 = 2 * AC ^ 2.

(24 * √2) ^ 2 = 2 * AC ^ 2.

AC ^ 2 = 576.

AC = 24 cm, then HM = KM = CK = CH = 24/2 = 12 cm.

Let’s define the perimeter of the square. P = 4 * KM = 4 * 12 = 48 cm.

Answer: The perimeter of the square is 48 cm.