Find the legs of a right-angled triangle if it is known that one of them is 7cm larger than the other

Find the legs of a right-angled triangle if it is known that one of them is 7cm larger than the other and the hypotenuse of this triangle is 13cm.

1) Suppose that x cm is the length of one leg of the triangle, then (x + 7) cm is the length of its second leg.

2) By the Pythagorean theorem in a right-angled triangle, the square of the hypotenuse is equal to the sum of the squares of the legs, so you can write:

x ^ 2 + (x + 7) ^ 2 = 13 ^ 2.

3) Let’s solve the equation:

x ^ 2 + x ^ 2 + 14x + 49 = 169;

2x ^ 2 + 14x + 49 – 169 = 0;

2x ^ 2 + 14x – 120 = 0;

x ^ 2 + 7x – 60 = 0.

By Vieta’s theorem:

x1 + x2 = -7,

x1 * x2 = -60, where x1 and x2 are the roots of the quadratic equation.

By selection, we find that x1 = -12, x2 = 5.

4) x1 = -12 is not a solution to the problem, since the length of the leg cannot be negative.

5) We get that x = 5 cm – the length of one leg.

6) 5 + 7 = 12 cm – the length of the second leg.

Answer: 5 and 12 cm.