# In a right-angled triangle, one leg is 3 cm less than the hypotenuse, and the other is 6 cm

**In a right-angled triangle, one leg is 3 cm less than the hypotenuse, and the other is 6 cm less than the hypotenuse. find the hypotenuse.**

Suppose that the hypotenuse of a given right-angled triangle is x cm.Then, according to the condition of the problem, one leg will be equal to x – 3 cm, and the second leg – x – 6 cm.

Let’s use the Pythagorean theorem and get the following equation:

(x – 3) ² + (x – 6) ² = x²,

x² – 6 * x + 9 + x² – 12 * x + 36 = x²,

x² – 18 * x + 45 = 0.

The discriminant of this quadratic equation is:

(-18) ² – 4 * 1 * 45 = 324 – 180 = 144.

So the equation has the following solutions:

x = (18 + 12) / 2 = 15 and x = (18 – 12) / 2 = 3.

Since x – 6 is a positive number, the solution to the problem

x = 15 (cm) – the length of the hypotenuse.