Stating and Applying Pythagoras' Theorem

The theorem states that a squared plus b squared equals c squared, where a and b are the two legs and c is always the hypotenuse. In words: the square on the hypotenuse equals the sum of the squares on the other two sides. The word 'always' in 'c is always the hypotenuse' is doing real work — putting a leg where c belongs breaks every step that follows. And there is one condition: this works only in right-angled triangles. If a question gives no right angle, Pythagoras is not the tool. Look for the little square in the diagram, or for words like vertical, perpendicular, upright, or due north, which imply one.

Here is the layout to use every time. Legs of 6 and 8, find the hypotenuse. Line one: write the theorem, a squared plus b squared equals c squared. Line two: substitute the legs, so 6 squared plus 8 squared equals c squared. Line three: square each number, giving 36 plus 64. Line four: add, giving 100 equals c squared. Line five: square root both sides, so c is 10. Five short lines, and you should write every one, because each carries a method mark. Jumping straight from 'six and eight' to 'ten' earns you the answer mark and nothing else — and if you slip, nothing at all.

The theorem also runs backwards, letting you test whether a triangle is right-angled at all. Take 9, 12, 15. The longest side is 15, so that is the only possible hypotenuse. Square the two shorter sides: 81 plus 144 is 225. Square the longest: 15 squared is also 225. They match, so yes, it is right-angled. Now take 5, 6, 8. The longest is 8, and 25 plus 36 is 61, while 8 squared is 64. Sixty-one is not sixty-four, so no. The key step is identifying the longest side first, because it is the only candidate for the hypotenuse — and always finish by saying what the numbers mean in words.

Most real triangles do not give whole-number answers. Legs of 5 and 7 give 25 plus 49, which is 74, so c is the square root of 74. That surd is the exact answer, and as a decimal it is about 8.60 to two decimal places. Read the question to see which form it wants: 'give an exact answer' means leave it as root 74, while 'to two decimal places' means round it. Then sense-check using square numbers you know: 64 is less than 74 which is less than 81, so root 74 must sit between 8 and 9 — and 8.60 does. It also beats the longest leg of 7 and stays under 5 plus 7.

To finish, here is what a full-mark solution looks like. Label the diagram with a, b and c. State the theorem before substituting. Show each squared value separately. Keep the exact value until the last line, round only if the question asks, write the units, and sense-check against the legs. The things to avoid are calling a leg c, adding before squaring, rounding partway through, and using Pythagoras where there is no right angle. But the single biggest mark-loser is forgetting the final square root — reaching 100 equals c squared and writing c equals 100. Your sense-check catches it instantly, because with legs of 6 and 8 the hypotenuse must be under 14.