In a right-angled triangle, the height drawn to the hypotenuse is 5 cm. Legs on the hypotenuse

In a right-angled triangle, the height drawn to the hypotenuse is 5 cm. Legs on the hypotenuse are related as 1: 25. Find these projections.

Let the larger leg of this triangle be a, and the smaller leg b.

If the projection of the smaller leg to the hypotenuse is x, then the projection of the larger leg is 25 * x. So the length of the hypotenuse is x + 25 * x = 26 * x.

Let us express the height of the triangle through its legs and their projections:

5² = a² – (25 * x) ²,

5² = b² – x².

From these equations we get:

a² = 625 * x² + 25,

b² = x² + 25.

By the Pythagorean theorem, we obtain the following equation:

a² + b² = (26 * x) ².

625 * х² + 25 + х² + 25 = 676 * х²,

626 * x² + 50 = 676 * x²,

50 * х² = 50,

x = 1.

Therefore, the projection of the larger leg is 1 * 25 = 25 cm, and the projection of the smaller leg is 1 * 1 = 1 cm.