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

Let’s denote by x the length of the shorter side of this rectangle.

According to the condition of the problem, one side of this rectangle is 14 cm larger than its other side, therefore, the length of the longer side of this rectangle is x + 14 cm.

It is also known that the diagonal of a rectangle is 34 cm, therefore, using the Pythagorean theorem, we can write:

x ^ 2 + (x + 14) ^ 2 = 34 ^ 2.

We solve the resulting 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 ± √529 = -7 ± 23;

x1 = -7 – 23 = -30;

x1 = -7 + 23 = 16.

Since the length of the side of the rectangle is positive, the value x = -30 is not suitable.

Find the length of the longer side of this rectangle:

x + 14 = 16 + 14 = 30 cm.

Find the area of ​​this rectangle:

30 * 16 = 480 cm ^ 2.

Answer: the area of ​​this rectangle is 480 cm ^ 2.