The perimeter of a right-angled triangle is 84 cm, and its nipotenuse is 37 cm. Find the area of this triangle.

Let’s denote the sides of a right-angled triangle a, b and c, where c is the hypotenuse, the two remaining sides are legs. Then you can compose an equation using the Pythagorean theorem:

c ^ 2 = a ^ 2 + b ^ 2 = 37 ^ 2 = 1369 cm2;

And if you use the perimeter formula, then you can make up the second equation:

P = a + b + c = a + b + 37 = 84 cm;

Let’s solve the resulting system of equations and find the length of the legs:

a + b + 37 = 84;

a + b = 84 – 37 = 47;

b = 47 – a;

a ^ 2 + b ^ 2 = 1369;

a ^ 2 + (47 – a) ^ 2 = 1369;

a ^ 2 + (472 – 2 * 47 * a + a ^ 2) = 1369;

a ^ 2 + 2209 – 94 * a + a ^ 2 = 1369;

2 * a ^ 2 – 94 * a + 2209 – 1369 = 0;

2 * a ^ 2 – 94 * a + 840 = 0;

2 * (a ^ 2 – 47 * a + 420) = 0;

a ^ 2 – 47 * a + 420 = 0;

a ^ 2 – (12 + 35) * a + (12 * 35) = 0;

Using Vieta’s theorem, we factor the equation and equate each to zero, since in the case of equality of the product to zero, at least one factor is equal to zero:

(a – 12) * (a – 35) = 0;

a – 12 = 0;

a = 12;

a – 35 = 0;

a = 35;

b = 47 – a = 47 – 35 = 12;

Knowing the legs, you can find the area of ​​a right-angled triangle, which is half the product of the legs:

S = a * b / 2 = 35 * 12/2 = 35 * 6 = 210 cm2.