The sum of the legs of a right-angled triangle is 49 m, and its hypotenuse is 41 m. Find the area of the triangle.

Let’s designate: x meters and y meters – the lengths of two legs.
According to the condition of the problem, a system of equations was compiled (the second equation was compiled according to the Pythagorean theorem):
x + y = 49;

x ^ 2 + y ^ 2 = 412;

Express x from the first equation and substitute it into the second:
x = 49 – y;

(49 – y) ^ 2 + y ^ 2 = 1681;

2401 – 98y + y ^ 2 + y ^ 2 = 1681;

2y ^ 2 – 98y + 720 = 0;

y ^ 2 – 49y + 360 = 0;

Discriminant = (-49) * (-49) – 4 * 360 = 961 (root of 961 is 31);

y = (49 + 31) / 2 or y = (49 – 31) / 2;

y = 40 or y = 9;

The x values ​​can take on the same values ​​as the y values. This means that the sides of the triangle are 40 meters and 9 meters. Let’s find the area of ​​a right-angled triangle:
40 * 9/2 = 180 m2;

Answer: the area of ​​the triangle is 180 m2.