In a right-angled triangle, one of the legs is 7cm larger than the other. Find the perimeter of the triangle if its hypotenuse is 13cm.
Let one leg of a right-angled triangle be x cm, then the second leg is (x + 7) cm.By the Pythagorean theorem in a right-angled triangle, the square of the hypotenuse is 13 ^ 2, and is equal to the sum of the squares of the legs (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 = 7 ^ 2 – 4 * 1 * (-60) = 49 + 240 = 289; √D = 17;
x = (-b ± √D) / (2a);
x1 = (-7 + 17) / 2 = 10/2 = 5 (cm) – the first leg;
x2 = (-7 – 17) / 2 = -24/2 = -12 – side length cannot be negative;
x + 7 = 5 + 7 = 12 (cm) – second leg.
Find the perimeter of the triangle. The perimeter of a triangle is equal to the sum of the lengths of its sides. P = a + b + c.
P = 5 + 12 + 13 = 30 (cm).
Answer. 30 cm.