Pythagoras' Theorem
For any right-angled triangle:
\(a^2 + b^2 = c^2\)
where&nbsp;\(a\), \(b\) are the lengths of the two shorter sides, and \(c\) is the hypotenuse.
Pythagorean Triples
•	A set of three positive integers \((a,b,c)\) that satisfy \(a^2 + b^2 = c^2\) are called Pythagorean Triples.
•	Examples: \((3,4,5)\), \((5,12,13)\), \((6,8,10)\), \((7,24,25)\), \((8,15,17)\), etc.
<ol class="abc-list"><li><div class="list-item-content">Generate Pythagorean Triples using an odd number (except \(1\)): Given any odd number, \(2n+1\), where&nbsp;\(n=1,2,3,...\), the square of \(2n+1\) can be expressed as the sum of two consecutive numbers, such that&nbsp;\((2n+1)^2 = k + (k+1)\). Using an algebraic method, or otherwise, show that&nbsp;\((2n+1, k, k+1)\) is a set of Pythagorean Triples.</div></li><li><div class="list-item-content">Generate Pythagorean Triples using an even number (except \(2\)): Given any even number, \(2n\), where&nbsp;\(n=2,3,...\), then the square of a half of \(2n\) is \(n^2\). Consider the other two numbers that are one less than and one more than \(n^2\), that is, \(n^2 - 1\) and \(n^2 + 1\). Using an algebraic method, or otherwise, show that&nbsp;\((2n, n^2 - 1, n^2 + 1)\) is a set of Pythagorean Triples.</div></li></ol>

where \(a\), \(b\) are the lengths of the two shorter sides, and \(c\) is the hypotenuse.

•	A set of three positive integers \((a,b,c)\) that satisfy \(a^2 + b^2 = c^2\) are called Pythagorean Triples.

•	Examples: \((3,4,5)\), \((5,12,13)\), \((6,8,10)\), \((7,24,25)\), \((8,15,17)\), etc.

Generate Pythagorean Triples using an odd number (except \(1\)): Given any odd number, \(2n+1\), where \(n=1,2,3,...\), the square of \(2n+1\) can be expressed as the sum of two consecutive numbers, such that \((2n+1)^2 = k + (k+1)\). Using an algebraic method, or otherwise, show that \((2n+1, k, k+1)\) is a set of Pythagorean Triples.

Generate Pythagorean Triples using an even number (except \(2\)): Given any even number, \(2n\), where \(n=2,3,...\), then the square of a half of \(2n\) is \(n^2\). Consider the other two numbers that are one less than and one more than \(n^2\), that is, \(n^2 - 1\) and \(n^2 + 1\). Using an algebraic method, or otherwise, show that \((2n, n^2 - 1, n^2 + 1)\) is a set of Pythagorean Triples.