Find the sides of a rectangle if one is 14 cm larger than the other, and the diagonal is exactly 34 cm.

Let x denote the length of the longer side of this rectangular quadrangle.

In the initial data for this task, it is reported that one side of this rectangular quadrangle is 14 centimeters larger than its other side, therefore, the length of the smaller side of this quadrangle is x – 14 cm.

Since the length of the diagonal of this quadrangle is 34 cm, using the Pythagorean theorem, we obtain the following equation:

x ^ 2 + (x – 14) ^ 2 = 34 ^ 2.

We solve this equation:

x ^ 2 + x ^ 2 – 28x + 196 = 1156.

2x ^ 2 – 28x + 196 – 1156 = 0;

2x ^ 2 – 28x – 960 = 0;

x ^ 2 – 14x – 480 = 0;

x = 7 ± √ (49 + 480) = 7 ± √ (49 + 480) = 7 ± √529 = 7 ± 23;

x = 7 + 23 = 30 cm.

Find the smaller side:

x – 14 = 30 – 14 = 16 cm.

Answer: the sides of the rectangle are 30 cm and 16 cm.