The hypotenuse of a right-angled triangle is 13 cm, and its perimeter is 30 cm. Find the legs of the triangle

Let’s denote by x the length of one of the legs of this right-angled triangle.

Let us express in terms of x the length of the second leg of this right-angled triangle.

According to the condition of the problem, the hypotenuse of this right-angled triangle is 13 cm, and its perimeter is 30 cm, therefore, the length of the second leg of this right-angled triangle is 30 – 13 – x = 17 – x.

Using the Pythagorean theorem, we get the following equation:

x ^ 2 + (17 – x) ^ 2 = 13 ^ 2.

We solve the resulting equation:

x ^ 2 + 289 – 34x + x ^ 2 = 168;

2x ^ 2 – 34x + 289 – 169 = 0;

2x ^ 2 – 34x + 120 = 0;

x ^ 2 – 17x + 60 = 0;

x = (17 ± √ (289 – 4 * 60)) / 2 = (17 ± √ (289 – 240)) / 2 = (17 ± √49) / 2 = (17 ± 7) / 2;

x1 = (17 + 7) / 2 = 24/2 = 12;

x2 = (17 – 7) / 2 = 10/2 = 5.

We find the second leg:

For x = 12:

17 – x = 17 – 12 = 5;

For x = 5:

17 – x = 17 – 5 = 12.

Answer: the legs of this right-angled triangle are 5 cm and 12 cm.