# In a right-angled triangle, one of the legs is 7 cm larger than the other. Find the perimeter of the triangle

In a right-angled triangle, one of the legs is 7 cm larger than the other. Find the perimeter of the triangle if its hypotenuse is 13 cm.

Let the first leg be x cm, then the second leg is (x + 7) cm.By the Pythagorean theorem: The square of the hypotenuse is equal to the sum of the squares of the legs, we have: the square of the hypotenuse is 13 ^ 2, the sum of the squares of the legs is x ^ 2 + (x + 7 ) ^ 2. Let’s make an equation and solve it.

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

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

2x ^ 2 + 14x + 49 – 169 = 0;

2x ^ 2 + 14x – 120 = 0;

x ^ 2 + 7x – 60 = 0;

D = b ^ 2 – 4ac;

D = 7 ^ 2 – 4 * 1 * (-60) = 49 + 240 = 289; √D = 17;

x = (-b ± √D) / (2a);

x1 = (-7 + 17) / 2 = 10/2 = 5 (cm) – 1st leg;

x2 = (-7 – 17) / 2 = -24/2 = -12 – length cannot be negative;

x + 7 = 5 + 7 = 12 (cm) – 2nd leg.

Find the perimeter of the triangle. The perimeter of a triangle is equal to the sum of the lengths of its sides.

P = 5 + 12 + 13 = 30 (cm).