# The length of the hypotenuse of the triangle is 37 cm, the total volume is 210 cm2. find the length of the leg.

Let “c” be the hypotenuse, “a” and “b” legs of the rectangle.

From the Pythagorean theorem we have that the square of the hypotenuse of a rectangle is equal to the sum of the squares of its legs: c ^ 2 = a ^ 2 + b ^ 2, and this is: 37 ^ 2 = a ^ 2 + b ^ 2.

We also have that the area of a right-angled triangle is calculated by the formula:

S = 1 / 2ab = 210.

The result is a system of equations: a ^ 2 + b ^ 2 = 37 ^ 2 and 1 / 2ab = 210.

We multiply the second equation by 4 and, in the first case, we subtract from the first equation, and in the second we add to the first, as a result we will have:

{a ^ 2 – 2ab + b ^ 2 = 37 ^ 2 – 840 and a ^ 2 + 2ab + b ^ 2 = 372 + 840} →

{(a – b) ^ 2 = 529 and (a + b) ^ 2 = 2209} → {(a – b) ^ 2 = 23 ^ 2 and (a + b) ^ 2 = 47 ^ 2} →

{(a – b) = 23 and (a + b) = 47} → {2a = 70 and 2b = 24} → {a = 35 and b = 12}.

Answer: 12 and 35.