The most general formulas for deriving all integer sided right-angled Pythagorean triangles, have been known since the days of Diophantus and the early Greeks. For a right triangle with sides X, Y, and Z, Z being the hypotenuse, the lengths of the three sides of the triangle can be derived as follows: X = k(m^2 - n^2), Y = k(2mn), and Z = k(m^2 + n^2) where k = 1 for primitive triangles (X, Y, and Z having no common factor), m and n are arbitrarily selected integers, one odd, one even, usually called generating numbers, with m greater than n. The symbol ^ means "raised to the power of" such that m^2 means m squared, etc.

Ftom your given information, using x and y as the two sides, you can write xy /2 = 180 and x^2 + y^2 = 41^2, 41 being the hypotenuse.

The 41 derives from z = k(m^2 + n^2).

Assuming k = 1, by inspection, 25 + 16 = 41 making m = 5 and n = 4.

The other two sides then become x = 5^2 - 4^2 = 9 and y = 2(5)4 = 40.

The area is therefore A 9(40)/2 = 180.

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