The difference between a right-angled triangle is 13 cm, find its legs if it is known that one of them is 7 cm

The difference between a right-angled triangle is 13 cm, find its legs if it is known that one of them is 7 cm larger than the other.

Let us denote by x the smaller leg of this right-angled triangle.

According to the condition of the problem, one of the legs of this right-angled triangle is 7 cm larger than the other, therefore, the length of the larger leg of this triangle is x + 7 cm.

It is also known that the length of the hypotenuse of a given right-angled triangle is 13 cm.

Using the Pythagorean theorem, we can compose the following equation:

x² + (x +7) ² = 13².

We solve the resulting equation:

x² + x² + 14x + 49 = 169;

2x² + 14x + 49 – 169 = 0;

2x² + 14x – 120 = 0;

x² + 7x – 60 = 0;

x = (-7 ± √ (7² + 4 * 60)) / 2 = (-7 ± √ (49 + 240)) / 2 = (-7 ± √289) / 2 = (-7 ± 17) / 2 ;

x1 = (-7 – 17) / 2 = -24 / 2 = -12;

x2 = (-7 + 17) / 2 = 10/2 = 5.

Since the length of the leg of a right-angled triangle is positive, the value x = -12 is not suitable.

Therefore, the length of the smaller leg of this right-angled triangle is 5 cm.

Knowing the smaller leg, we find the larger one:

x + 7 = 5 7 = 12 cm.

Answer: the legs of this right-angled triangle are 5 cm and 12 cm.