One side of the rectangle is 7 cm shorter than the other, and its diagonal is 13 cm. Find the sides of the rectangle.

Suppose that one of the sides of the rectangle is x cm, then the other side will be x + 7 cm.

The diagonal of the rectangle and its sides make up a right-angled triangle, the hypotenuse of which is equal to the diagonal, and the legs are equal to the sides of the rectangle.

Let’s use the Pythagorean theorem and compose the equation:

x² + (x + 7) ² = 13²,

x² + x² + 14 * x + 49 = 169,

2 * x² + 14 * x – 120 = 0,

x² + 7 * x – 60 = 0.

The discriminant of this equation is:

7² – 4 * 1 * (-60) = 49 + 240 = 289.

Since x can only be a positive number, the problem has a unique solution:

x = (- 7 + 17) / 2 = 5 (cm) – the length of one side of the rectangle.

17 – 5 = 12 (cm) – the length of the second side of the rectangle.