Finding a Shorter Side

Now the other kind of question: the hypotenuse is known and one leg is missing. With hypotenuse 13 and one leg 5, write a squared plus 5 squared equals 13 squared — the hypotenuse always keeps its place as c. That gives a squared plus 25 equals 169, so a squared is 169 minus 25, which is 144, and a is 12. The rule is that finding a leg means subtracting, because you know the big square and one small square and you want the other small one. Sense-check: a leg must be smaller than the hypotenuse, and 12 is less than 13. If your leg came out bigger than the hypotenuse, you added when you should have subtracted.

Everything in this chapter comes down to one question, asked before you write anything: is the missing side the hypotenuse? If yes, both legs were given, so you add, and your answer will be the biggest side — bigger than either number you started with. If no, the hypotenuse was given, so you subtract, and your answer will be a shorter side, smaller than the hypotenuse. A quick practical test is to look at where the biggest number is. If the biggest number was given to you, subtract. If the biggest number is what you are being asked for, add. Ask this first and the arithmetic looks after itself.

The ladder problem is the classic version of this. A 5 metre ladder has its foot 1.4 metres from a wall — how high does it reach? The ladder is the hypotenuse, because it leans from the ground to the wall, opposite the right angle at the base. So h squared plus 1.96 equals 25, which gives h squared equals 23.04, and h is 4.8 metres. Less than the 5 metre ladder, as it must be. Because the ladder is always the hypotenuse, ladder questions are nearly always subtract questions, unless the ladder's own length is what is missing. Decimals are fine here — just square them carefully, since 1.4 squared is 1.96, not 2.8.

Isosceles triangles come up constantly, and they need one extra move. With equal sides of 13 and a base of 10, drop the height from the apex — it splits the base exactly in half, making two identical right-angled triangles. Work with just one of them: half the base is 5, and the equal side of 13 becomes that half's hypotenuse. So h squared plus 25 equals 169, h squared is 144, and h is 12. The trap is using the whole base of 10 instead of the half. That would give 169 minus 100, or 69, and a height around 8.3 — completely wrong. Halve the base before substituting, and write 'half base equals 5' on your page so you cannot forget.

Finish every question with three checks that take about ten seconds. First, size: is your leg smaller than the hypotenuse? A leg of 15 with a hypotenuse of 13 is impossible. Second, substitute back: take your three sides and test whether a squared plus b squared really equals c squared — for 5, 12 and 13, 25 plus 144 is 169 ✓. This one is the powerful check, because it catches add-and-subtract mix-ups that nothing else will. Third, units and question: did you answer what was asked, and write the units? '4.8 metres', not a bare '4.8'. Ten seconds of checking is worth more than ten minutes of extra practice.