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

Let us denote by a 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 7 centimeters larger than its other side, therefore, the length of the smaller side of this quadrangle is a – 7 cm.

Since the length of the diagonal of this quadrangle is 13 cm, using the Pythagorean theorem, we obtain the following equation:

a ^ 2 + (a – 7) ^ 2 = 13 ^ 2.

We solve this equation:

a ^ 2 + a ^ 2 – 14a + 49 = 169;

2a ^ 2 – 14a + 49 – 169 = 0;

2a ^ 2 – 14a – 120 = 0;

a ^ 2 – 7a – 60 = 0;

a = (7 ± √ (49 + 4 * 60)) / 2 = (7 ± √289) / 2 = (7 ± 17) / 2;

a = (7 + 17) / 2 = 24/2 = 12 cm.

Find the smaller side:

a – 7 = 12 – 7 = 5 cm.

Answer: the sides of the rectangle are 12 cm and 5 cm.