The perimeter of the rectangular hall is 34 cm, and the length of the diagonal is 13 cm. Calculate the sides of the hall.

Let x denote the length of this rectangular hall.

The wording of the condition for this task states that the sum of the lengths of all sides of this hall is 34 meters.

Consequently, the half-sum of the lengths of all sides of this hall is 34/2 = 17 meters and the width of this hall should be equal to 17 meters.

Since the length of the diagonal of the hall is 13 meters, using the Pythagorean theorem, we obtain the following equation:

x ^ 2 + (17 – x) ^ 2 = 13 ^ 2,

solving which, we get:

x ^ 2 + 289 – 34x + x ^ 2 = 169;

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.

Find the second side:

17 – x1 = 17 – 12 = 5;

17 – x2 = 17 – 5 = 12.

Answer: 5 meters and 12 meters.