Right-angled Triangles and the Hypotenuse

Before any calculating, you need to name the sides correctly. In a right-angled triangle the hypotenuse is the side opposite the right angle — it is always the longest side, and it never touches the right angle. The other two sides are called the legs, and they are the two that form the right angle. Here is the part students get wrong: the hypotenuse is not 'the slanted one' or 'the one at the bottom'. Triangles get rotated, and a diagram can put the right angle anywhere on the page. So find the little square marking the right angle first, every single time, then look straight across from it.

Why must the hypotenuse be the longest side? Think about walking from one corner to another. Going along the two legs of a 3-4 triangle is 3 plus 4, which is 7 units of walking. Cutting straight across the hypotenuse is only 5. The direct route is always shorter than going around the corner, but it can never be shorter than either leg on its own. That gives you a free sense-check on every answer you ever calculate: the hypotenuse must be bigger than each leg but smaller than their sum. For legs of 3 and 4, your answer has to land between 4 and 7 — and 5 does. An answer of 2 or of 12 would be wrong before you even check the arithmetic.

Here is the idea the whole chapter rests on. Take a right-angled triangle with legs 3 and 4 and a hypotenuse of 5, and build an actual square on each side. The square on the leg of 3 has area 9. The square on the leg of 4 has area 16. And the square on the hypotenuse has area 25. Now notice: 9 plus 16 equals 25. The two small squares together hold exactly the same area as the big one. That is Pythagoras' theorem in one picture, and it is why the formula deals in squared numbers rather than the side lengths themselves. It is a statement about areas, not about lengths.

Pythagoras needs two operations. Squaring: 7 squared means 7 times 7, which is 49 — not 7 times 2. That little raised 2 is not a multiplier, and treating it as one is the most common mistake in the whole chapter. Square rooting undoes squaring: the square root of 49 is 7, because 7 times 7 makes 49. It is worth learning the square numbers by heart up to about 400, because recognising 169 as 13 times 13 saves reaching for a calculator. More importantly it tells you whether an answer is sensible — the square root of 150 has to sit between 12 and 13, since 144 is less than 150 which is less than 169.

Diagrams will rotate the triangle to test whether you actually understand the naming. In all three of these the white dot marks the right angle and the gold side is the hypotenuse — but it sits in a different place each time. So build yourself a routine and run it on every question: find the right angle, mark the hypotenuse opposite it, label the two legs, and then decide whether you are being asked for the hypotenuse or for a leg. That last step matters most, because it decides whether you will add or subtract in the next lesson. Two minutes labelling a diagram saves you from the error that costs the most marks.