Deriving the distance rule by hanging a right triangle off the two points

0:00Two towns. The second one is thirty-six kilometres east of the first and fifteen kilometres north of it. How far apart are they? Not thirty-six plus fifteen. That is the distance you walk going east first and then north, and nothing travels in an L. Not thirty-six either, and not fifteen. Those are the two legs of the walk, not the journey. The straight line between the towns is shorter than the walk and longer than either leg, and so far we have no way at all to say what it is.

0:37What we do have is two numbers laid at right angles to each other, which is the shape of every coordinate pair there has ever been. So this is not really a question about towns. It is a question about what two readings can tell you about a length. By the end there will be one rule that answers it, and the rule will turn out to contain no new mathematics whatsoever.

1:04Start with the separations we can already do. Two points on the horizontal ruler, one at four across and one at six across. Six take away four. Two units. Two points on the vertical ruler, at three up and at eight up. Eight take away three. Five units. The reason those work is not that the points happen to sit on a ruler. It is that they agree in their other reading. Both of the first pair are at height zero, so the whole gap lies along one direction.

1:39And when a gap lies along one direction, a single subtraction catches all of it. Take a grid of eighty points and look at every pair. Six hundred and forty pairs share a reading and subtracting handles them. On the other two thousand five hundred and twenty it fails every time. So here is a pair that does not cooperate. P sits at four across and six up. Q sits at six across and eight up.

2:12Subtract the first readings: six take away four is two. Subtract the second readings: eight take away six is two. Two and two. Two what? Neither of those numbers is the separation. They are the two ways in which the points differ, and the separation is some third thing built out of both. You can see it in the picture. P and Q sit on a slant, and the slant is plainly longer than two.

2:42But there is no ruler laid along that slant. Nothing in the diagram has measured it, and nothing is going to. So we are going to have to build something that has. Here is the move, and it is the only inventive step in this entire topic. Drop a perpendicular from P straight down to the horizontal ruler. Drop a second one from Q. Now run a horizontal line from P across until it meets that second perpendicular. Call the meeting point T.

3:18Look at what has appeared. P, T and Q are the three corners of a triangle, and the corner at T is a right angle. Nothing in the question contained a right angle. Two points and the slant between them have no angle in them at all. We put one there. Three lines that were not in the problem, drawn for no reason except that a right angle is a thing we already know how to handle.

3:46That is what a construction is. Not a discovery. A manufacture. Now the part that makes the whole manoeuvre worth doing. The two new sides never have to be measured, because the rulers measured them already. PT runs horizontally. Its two ends are at four across and six across, and the gap between two points on one horizontal line is the difference of their first readings. Two. TQ runs vertically. Its ends are at six up and eight up. The difference of the second readings. Also two.

4:27Each leg is a coordinate difference. Not a length you squint at on a diagram, but an arithmetic you do on the numbers you were handed at the start. That is the second half of the trick. The construction makes a right triangle, and the coordinates fill in its two legs for nothing. Across two thousand five hundred and twenty pairs of points, every single time, both legs come out as differences and neither one is ever measured.

4:59A right triangle with two known legs has a known third side. That is the oldest tool in the box. Square the first leg: two squared is four. Square the second: also four. Add them: eight. That is the square of the separation. Then take the square root. P and Q are the root of eight apart. And that is the whole rule. Square the difference of the first readings, square the difference of the second, add the two, take the root.

5:32No new theorem entered anywhere in that. The only thing that happened is that a right triangle got built where there had not been one. Which means every separation you ever compute this way is Pythagoras, wearing coordinates. Try it somewhere the numbers are less friendly. P at six across and four up. Q five to the left and three down, so its two readings are minus five and minus three.

6:03Same construction, line for line. Drop a perpendicular from Q to the horizontal ruler, run a horizontal from P across to meet it, and mark the corner. That corner sits at minus five across and four up, because it is below Q and level with P. The horizontal leg runs from minus five across to six across. Its length is six take away minus five. Eleven. The vertical leg runs from minus three up to four up. Four take away minus three. Seven.

6:40Eleven squared is a hundred and twenty-one, seven squared is forty-nine, and they add to a hundred and seventy. The separation is the root of a hundred and seventy, and nothing about the method changed. But something in that example is worth stopping on, because it hides a mistake. Eleven is six take away minus five. Eleven is also six plus five. So on that figure you can get the right answer by ADDING the two distances from the ruler instead of subtracting the two readings, and nobody would ever catch you.

7:21It works there because the two points sit on opposite sides. Adding two distances and subtracting two readings are the same arithmetic only when you cross zero between them. Move one point over so that both are on the same side and they part company immediately. Six and two: subtracting gives four, adding gives eight. Over every pair in the grid of eighty, the adding rule is right on one thousand eight hundred and eighty-four of them and wrong on one thousand two hundred and seventy-six.

7:59And there is a cleaner reason it cannot be a length. Slide the entire picture a hundred units in each direction: every real separation is untouched, and the adding rule's answer changes on every single pair. A length does not know where you put the origin. That one does. Now run the identical construction with nothing but letters in it. P has readings x-one and y-one. Q has x-two and y-two. Drop the two perpendiculars, run the horizontal across, mark the corner.

8:38The corner sits directly below Q and directly across from P, so its readings are x-two and y-one. The horizontal leg is therefore x-two minus x-one, and the vertical leg is y-two minus y-one. Same argument, no numbers. Square them. Add them. Take the root. The separation of P and Q is the square root of x-two minus x-one all squared, plus y-two minus y-one all squared. That is the rule, and you have just watched it get built instead of being handed it.

9:19Two loose ends, and both of them are about squaring. Squaring throws away a sign, so at the last step there are two numbers whose square is what we computed. One positive, one negative. We keep the non-negative one, and not because somebody decided to. A separation is a size, and a size cannot be less than nothing. The second loose end is which point you called first. Swap them and every single difference changes sign.

9:51But every difference goes into the rule squared, and a squared number cannot tell you what sign it used to have. So the two orders agree, and they agree term by term rather than by some cancellation in the total. On three thousand one hundred and sixty pairs of points, run both ways round, the rule does not once return a different answer. Two more things fall out at no extra cost, and neither is a new rule.

10:23Put one of the two points at the origin. Both of its readings are nought, so both differences collapse to the other point's own readings. The separation of a point from the origin is the root of its first reading squared plus its second reading squared. That is not a second formula to memorise. It is the same one with two zeros substituted in, and the substituting is the part you should actually do.

10:53The other one is the swap we just made. Written out with the two points interchanged, the rule looks like a different expression. It is not. x-one minus x-two is the negative of x-two minus x-one, and squaring destroys every trace of the difference between them. Term by term, not by coincidence in the sum. Which is a thing worth checking rather than believing. One habit left, and it is worth arguing for rather than assuming.

11:28The first example came out as the root of eight. That is a correct answer and a fairly useless one, because nobody has any feel for how big the root of eight is. Eight is four times two, and four is a perfect square, so the root of eight is two roots of two. The root of two is about one point four, so the separation is about two point eight. Now you know its size.

11:55Same move on the root of fifty. Fifty is twenty-five times two, so it is five roots of two. But the root of a hundred and seventy has nothing to pull out. Two, five and seventeen multiply to give it, and not one of them appears twice. So it stays exactly as it is. Simplifying a root is not a ritual you perform on every answer. It is something you do when there is a square factor sitting there to be taken.

12:29The rule also runs backwards, and there it behaves differently in a way that is worth watching closely. P sits at two across and three down. Q sits at ten across, at some height we do not know. The two are ten units apart. What is the height? The horizontal difference is eight. Eight squared is sixty-four, the whole separation squared is a hundred, so the vertical difference squared has to be thirty-six.

13:01The vertical difference is six. Or it is minus six. And this time we keep both of them. Q is either six above P or six below. Its second reading is three, or it is minus nine, and putting either one back into the rule returns a hundred. Now compare that with the last step of the rule itself, where we deliberately threw one root away. There the unknown was a length, and a length has no sign to lose. Here the unknown is a coordinate, and a coordinate does. Same equation, different question, different number of answers.

13:43Back to where we started. One town, and the other thirty-six kilometres east of it and fifteen north. Put the first at the origin and the second one's readings are thirty-six and fifteen. Thirty-six squared is one thousand two hundred and ninety-six. Fifteen squared is two hundred and twenty-five. Those add to one thousand five hundred and twenty-one, and one thousand five hundred and twenty-one is thirty-nine squared, exactly. Thirty-nine kilometres. Not fifty-one, which is the walk, and not thirty-six, which is only the eastward part of it.

14:25Look back at what actually did the work there. Not the formula. The three lines nobody asked for. There was no right angle, and then there was one, and everything after that was arithmetic we already had.