The hypotenuse of a right-angled triangle is 15 cm and one of the legs is 3 cm smaller than the other.

The hypotenuse of a right-angled triangle is 15 cm and one of the legs is 3 cm smaller than the other. Find the legs of the triangle.

Let’s denote by x the length of the larger leg of this right-angled triangle.

According to the condition of the problem, one of the legs of this right-angled triangle is 3 cm smaller than the other, therefore, the length of the smaller leg of this right-angled triangle is x – 3 cm.

It is also known that the hypotenuse of this right-angled triangle is 15 cm, therefore, using the Pythagorean theorem, we can compose the following equation:

x ^ 2 + (x – 3) ^ 2 = 15 ^ 2.

We solve the resulting equation:

x ^ 2 + x ^ 2 – 6x + 9 = 225;

2x ^ 2 – 6x + 9 – 225 = 0;

2x ^ 2 – 6x – 216 = 0;

x ^ 2 – 3x – 108 = 0;

x = (3 ± √ (9 + 4 * 108)) / 2 = (3 ± √ (9 + 4 * 108)) / 2 = (3 ± √441) / 2 = (3 ± 21) / 2;

x = (3 + 21) / 2 = 24/2 = 12.

We find the second leg:

x – 3 = 12 – 3 = 9.

Answer: the legs of this right-angled triangle are 9 cm and 12 cm.