The perimeter of a right-angled triangle is 30 cm, the length of the hypotenuse is 13 cm. Find the area of this triangle.

The area of ​​a right-angled triangle is half the product of its legs, so we need to find the legs of the triangle. If the perimeter of 30 cm and the hypotenuse are known. then the sum of the two legs is 30 – 13 = 17 (cm).

Let one leg be x cm, then the second leg is (17 – x) cm.By the Pythagorean theorem, we compose an equation and solve it.

13 ^ 2 = x ^ 2 + (17 – x) ^ 2 – open the bracket according to the formula for the square of the difference of two expressions;

169 = x ^ 2 + 289 – 34x + x ^ 2;

2x ^ 2 – 34x + 120 = 0 – divide by 2;

x ^ 2 – 17x + 60 = 0;

D = b ^ 2 – 4ac;

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

x = (- b ± √D) / (2a)

x1 = (17 + 7) / 2 = 24/2 = 12 (cm) – length of the first leg, 17 – 12 = 5 (cm) – length of the second leg;

x2 = (17 – 7) / 2 = 10/2 = 5 (cm) – the length of the first leg, 17 – 5 = 12 (cm) – the length of the second leg.

S = 1/2 * 12 * 5 = 6 * 5 = 30 (cm ^ 2).

Answer. 30 cm ^ 2.