The hypotenuse of a right-angled triangle is 13 cm, and one of the legs is 7 cm larger than the other.

The hypotenuse of a right-angled triangle is 13 cm, and one of the legs is 7 cm larger than the other. Find the sides of the triangle.

Let’s denote the first leg of a right-angled triangle – x cm, then the second will be equal to (x – 7) cm.

According to the Pythagorean theorem, in a right-angled triangle, the sum of the squares of the legs is equal to the square of the hypotenuse. Let’s make an equation and solve it.

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

x ^ 2 + (x -7) ^ 2 = 169.

x ^ 2 + (x ^ 2 – 14x + 49) = 169.

x ^ 2 + x ^ 2 – 14x + 49 = 169.

2x ^ 2 – 14x + 49 – 169 = 0.

2x ^ 2 – 14x – 120 = 0.

Divide the equation by 2 and solve the resulting quadratic equation.

x ^ 2 – 7x – 60 = 0.

Let’s find the discriminant.

D = b ^ 2 – 4 * a * c = 49 – 4 * 1 * (- 60) = 49 + 240 = 249.

The solution has two roots.

By Vieta’s theorem.

x1 + x2 = 7,

x1 * x2 = – 60.

x1 = – 5,

x2 = 12.

Since the leg cannot have a negative length, the length of the first is 12 cm, and the length of the second is 12 – 7 = 5 cm.

Answer: The legs are 12 cm and 5 cm.