Pythagorean Triples

Must know: 3-4-5, 5-12-13, 8-15-17, 7-24-25. Worth knowing: 20-21-29, 9-40-41.

Odd shortest side n: the other sides are (n² − 1)/2 and (n² + 1)/2. 3 → 4, 5 · 5 → 12, 13 · 7 → 24, 25 · 9 → 40, 41 · 11 → 60, 61.
General formula: m² − n², 2mn, m² + n² (m > n). m=2,n=1 → 3-4-5 · m=3,n=2 → 5-12-13 · m=4,n=1 → 15-8-17 · m=5,n=2 → 21-20-29.

Spotting scaled triples: divide the given sides by a common factor. 45, 60, ? → ÷15 → 3, 4 → 5 → ×15 = 75. For decimals, multiply up first: 0.6, 0.8 → ×10 → 6, 8 = 2 × (3, 4) → hypotenuse 1.0.
Not a triple: 2-3-4, 4-5-6, 6-8-11 etc. Always check a² + b² = c² if unsure.