The perimeter of the right triangle is 56 cm. The hypotenuse is 25 cm. Find the legs of the right triangle.

Let us denote the lengths of the legs of this right-angled triangle through x and y.

In the condition of the problem it is said that the perimeter of this right-angled triangle is 56 cm, and its hypotenuse is 25 cm, therefore, the sum of the legs is:

x + y = 56 – 25 = 31.

From this ratio we get:

y = 31 – x.

Using the Pythagorean theorem, we get the following equation:

x ^ 2 + (31 – x) ^ 2 = 25 ^ 2.

We solve the resulting equation:

x ^ 2 + 961 – 62x + x ^ 2 = 625;

2x ^ 2 – 62x + 961 – 625 = 0;

2x ^ 2 – 62x + 336 = 0;

x ^ 2 – 31x + 168 = 0;

x = (31 ± √ (961 – 4 * 168)) / 2 = (31 ± √289) / 2 = (31 ± 17) / 2;

x1 = (31 + 17) / 2 = 24;

x2 = (31 – 17) / 2 = 7.

We find at:

y1 = 31 – x1 = 31 – 24 = 7;

y2 = 31 – x2 = 31 – 7 = 24.

Answer: the legs of a right-angled triangle are 7 cm and 24 cm.