Show that the area of a Pythagorean right triangle is aninteger. Note that a Pyt

Question

Show that the area of a Pythagorean right triangle is aninteger.

Note that a Pythagorean right triangle is a triangle such that (thelength of) all sides are integers and the normal right triangleproperties hold (Pythagorean Theorem, etc).

Since the area is 1/2*a*b where a and b are the sides that aren'tthe hypotenuse (equivalenly: ½*base*height) it suffices toshow that a*b is even....

Use any method(s) you want to prove it...modular arithmetic,etc...all fair game....

Thank you in advance for any help at all.

Explanation / Answer

If c is the hypotenuse, then c =(a2+b2) is an integer. which means (a2+b2) is a perfect square. Now, if (a2+b2) is an odd perfect square,then either a or b is even, right ? And, if (a2+b2) is an even perfect square,then it is multiple of 4 and which means either both a and b areeven or both a and b are odd. But if both a and b are odd then the sum of their squares can neverbe multiple of 4, so this possibility is ruled out. As a proof forthis, consider a to be 2m +1 and b to be 2n +1 and calculate(a2+b2), you will see this is neverpossible. So we find that for a Pythagorean triangle at least one of a and bmust be even, which in turn means that ab must be even.

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