Pythagorean Triads and Practical Applications

A Pythagorean triad is a set of three whole numbers that satisfy the theorem, and the largest is always the hypotenuse. The ones worth memorising are 3-4-5, 5-12-13, 8-15-17, 7-24-25, 9-40-41 and 20-21-29. Exam writers use them constantly, because they let a question have a tidy whole-number answer. Recognising 5, 12, 13 saves you the entire calculation. More usefully, it warns you when something has gone wrong — if a question gives you 5 and 12 and you get anything other than 13, you know to look again before you have even checked your working.

Triads keep working when you scale them. Double 3-4-5 and you get 6-8-10; triple it and you get 9-12-15; multiply by ten and you get 30-40-50. Check the middle one: 81 plus 144 is 225, which is 15 squared ✓. It works because multiplying every side by the same number keeps the triangle the same shape, just bigger. The real skill is spotting a multiple. If a question gives you legs of 30 and 40, think '3-4-5 times 10' and the answer is 50 with no working at all. Likewise 20 and 48 is 5-12-13 times 4, giving 52.

Here is the theorem doing real work. To check that a corner is square, a builder marks 3 metres up one wall and 4 metres along the other, then measures between the marks. Exactly 5 metres means the corner is a true right angle. More than 5 means it is too open, less than 5 means too closed. This is the theorem running backwards: if three sides satisfy the equation, the angle must be 90 degrees. Builders use 3-4-5, or 6-8-10 for bigger jobs, to square up foundations, decks, fences and tiled floors — no protractor required. It is around two thousand years old and still in use on site this morning.

In real problems Pythagoras is usually one step inside something bigger. A yard 12 by 9 metres needs fencing round the edges plus a diagonal path, at 18 dollars a metre. Step one: the diagonal, 144 plus 81 is 225, so 15 metres. Step two: the perimeter, 2 times 21, which is 42 metres. Step three: the total, 42 plus 15, which is 57 metres. Step four: the cost, 57 times 18, which is 1026 dollars. Break these down by asking what you need first and what that lets you find next. Label each step on your page, because examiners award marks per step — a wrong final answer with clear steps still earns most of them.

Here is the whole chapter as a checklist. One: is there a right angle? If not, this is not a Pythagoras question. Two: draw and label, with c opposite the right angle. Three: is the missing side the hypotenuse — yes means add, no means subtract. Four: do you recognise a triad or a multiple of one? Five: substitute, square each number, then combine. Six: take the square root. Seven: sense-check, add units, and re-read the question in case finding the hypotenuse was only step one. Steps three and six are where nearly all the marks are lost, so slow down on those two.