Pythagoras' theorem in two dimensions

Picture a builder setting out the corner of a new house. She puts one peg three metres along the line of one wall, and another peg four metres along the other. Then she measures straight across, from peg to peg. If that diagonal is exactly five metres, the corner is a perfect right angle. Three add four is seven, yet the tape has to read five. The reason is a rule about every right-angled triangle, and it's the rule this video teaches.

A right-angled triangle is any triangle with one square corner, a ninety-degree angle, usually marked with a small square. The side facing that square corner has its own name: the hypotenuse, which is the side stretched across from the right angle. It is always the longest side of the triangle, because it faces the biggest angle. To find it in any right-angled triangle, stand in the square corner and look straight across. The side you are looking at is the hypotenuse. Here is a triangle tipped over at an angle, with its three sides labelled p, q and r. Which side is the hypotenuse here: p, q or r? It's r. From the square corner, looking straight across, r is the side in front of you, however the triangle is turned. Now the rule itself, Pythagoras' theorem, which links the lengths of all three sides. Call the two shorter sides a and b, and call the hypotenuse c. The theorem says: a squared plus b squared equals c squared. In words, square each shorter side, add the two results together, and you get the hypotenuse squared.

Every Pythagoras question starts with one decision, made before any number goes in. Is the missing side the longest one? If it is, the hypotenuse is what you are hunting for. Square the two sides you know, add them, then square root. If it isn't, you already have c. Square the two sides you know, take the smaller square from the bigger one, then square root. Now try one: hypotenuse ten centimetres, one shorter side six. Add or subtract? Subtract, because the ten centimetres is the hypotenuse, facing the square corner. Ten squared is a hundred. Six squared is thirty-six. A hundred take away thirty-six leaves sixty-four. The square root of sixty-four is eight, so the missing side is eight centimetres. The subtraction comes straight out of the rule. If a squared plus b squared makes c squared, then c squared take away b squared leaves a squared. The rule also hands you a check. A hypotenuse always comes out longer than both other sides, and a shorter side always comes out shorter than the hypotenuse.

Now a real one. A zip-wire is fixed between two posts, and one end is nine metres higher than the other. The posts stand twelve metres apart. The nine metres of height and the twelve metres of gap meet at a right angle, and the wire joins their two ends. Is the wire the hypotenuse, or one of the shorter sides? The hypotenuse. It faces the square corner, and it's the side we're missing, so this time we add. Remember, a squared plus b squared equals c squared. Here that's nine squared plus twelve squared equals c squared. Nine squared is eighty-one. Twelve squared is a hundred and forty-four. Next, add them. Eighty-one plus a hundred and forty-four is two hundred and twenty-five, which makes c squared two hundred and twenty-five. The last step undoes the squaring. The square root of two hundred and twenty-five is fifteen, so the zip-wire is fifteen metres long, longer than both other sides, as a hypotenuse has to be. Here is the picture behind the rule. Build a square on each side of this triangle, and they hold eighty-one, a hundred and forty-four and two hundred and twenty-five unit squares. The two smaller squares fill the big one exactly. One examiner's report, on a question asking for a hypotenuse, says this. "Most students did not realise that they needed to use Pythagoras' theorem to work out the hypotenuse length of the right-angled triangle. Many simply added thirty-two and sixty to achieve an answer of ninety-two centimetres." What went wrong to give ninety-two centimetres? Nobody squared anything. The two sides went in as bare numbers, and sides never combine that way. The right method squares first. Thirty-two squared is one thousand and twenty-four. Sixty squared is three thousand six hundred. Add them to get four thousand six hundred and twenty-four. Its square root is sixty-eight, so the hypotenuse is sixty-eight centimetres. The fix is one habit: square every side before you add anything, and take the square root only at the very end.

Now the reverse decision. A builder's ramp is thirteen metres long, and it reaches a platform five metres above the ground. The question is how far along the ground its base runs. Line A is thirteen plus five. Line B is thirteen squared plus five squared. Line C is thirteen squared take away five squared. Which first line is right: A, B or C? C. The ramp faces the square corner, so it's the hypotenuse, and the missing side is a shorter one. Line A never squares, and line B only works when the hypotenuse is missing. That means we subtract. Remember, c squared take away b squared leaves a squared. Here that's thirteen squared take away five squared. Thirteen squared is a hundred and sixty-nine. Five squared is twenty-five. Next, subtract. A hundred and sixty-nine take away twenty-five leaves a hundred and forty-four. Then the square root. The square root of a hundred and forty-four is twelve, so the base of the ramp runs twelve metres along the ground. Twelve metres is shorter than the thirteen-metre ramp, as a shorter side must be. Line B adds instead: a hundred and sixty-nine plus twenty-five is a hundred and ninety-four. Its square root is about thirteen point nine metres, longer than the ramp itself. Another line in that same report names the opposite slip. "a common misconception was to incorrectly apply Pythagoras' theorem and subtract the squares of the sides to incorrectly achieve the square root of two thousand five hundred and seventy-six equals fifty point eight centimetres." Why can't fifty point eight centimetres be the hypotenuse? It's shorter than the side of sixty, and the side facing the right angle is always the longest. That two thousand five hundred and seventy-six is sixty squared take away thirty-two squared. So on the same triangle, the squares were taken away when they should have been added. The habit that stops this slip is the decision from the start: ask whether the missing side is the longest before a single number goes in.

Pythagoras also runs backwards, and that is the builder's trick with the tape measure. If the two shorter sides squared add up to the longest side squared, the triangle has a right angle. If they don't add up, it hasn't. For the builder's pegs, three squared is nine and four squared is sixteen. Nine plus sixteen is twenty-five. Five squared is twenty-five as well. The totals match, so the corner is square, and that's why the tape has to read five metres, not seven. Next: pegs at six and eight metres, diagonal eleven. Is that corner square? No. Six squared is thirty-six and eight squared is sixty-four, which add to a hundred. Eleven squared is a hundred and twenty-one. They differ, so it isn't a right angle. One last skill is spotting a Pythagoras question in the first place. Two sides known, no angle anywhere, and a length wanted. If the question gives you an angle as well, it needs trigonometry, which has its own videos in this chapter. If it asks for the space inside a shape, it wants area, not a length. Carry this through the whole chapter: every angle faces its own side - look straight across. From the square corner, the side you see is the hypotenuse, the longest side.

Three questions now, to see what has stuck from the ramp, the zip-wire and the builder's pegs. What's the one question to ask before any number goes in? Before any number: is the missing side the longest? If yes, add the squares. If no, subtract. Try this. Shorter sides eight and fifteen centimetres: how long is the hypotenuse? Seventeen centimetres. Eight squared is sixty-four and fifteen squared is two hundred and twenty-five. Together they make two hundred and eighty-nine, whose square root is seventeen. Now a different corner: pegs at twelve and sixteen metres, diagonal twenty. Square? Yes. A hundred and forty-four plus two hundred and fifty-six makes four hundred, and twenty times twenty is four hundred as well. That corner is a right angle.

If you could teach this one to a friend, the thumbs-up is yours to press. That way your list shows the videos you never need to watch again. If it hasn't landed yet, save it to a playlist for later. Three questions in, it starts to feel normal.

Next in the chapter: Trigonometric ratios, labelling sides and choosing sin, cos or tan.

For more, visit scholafly.com, or watch the next video.