The legs of a right-angled triangle are 5:12, and the median to the hypotenuse is 26 cm. Find the perimeter of the triangle.

Since in a right-angled triangle, the length of the median drawn to the hypotenuse is equal to half the length of the hypotenuse itself, then AC = 2 * BM = 2 * 26 = 52 cm.

Let the leg AB = 5 * X, then the leg BC = 12 * X.

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

52 ^ 2 = (5 * X) ^ 2 + (12 * X) ^ 2.

2704 = 25 * X ^ 2 + 144 * X ^ 2.

169 * X ^ 2 = 2704.

X ^ 2 = 2704/159 = 16.

X = 4. Then AB = 5 * 4 = 20 cm, BC = 12 * 4 = 48 cm.

Determine the perimeter of the triangle: Ravs = AB + BC + AC = 20 + 48 + 52 = 120 cm.

Answer: The perimeter of the triangle is 120 cm.