Constructing a rectangle from one side and a diagonal

0:00Here is a job. Build a rectangle. But nobody is going to tell you how long its sides are. All you get is this. Draw the line joining two opposite corners, the diagonal, and it leaves the bottom side at an angle of sixty degrees. That is the whole instruction. Sixty degrees. At first that sounds like far too little to go on. A rectangle has four corners and two side lengths, and you have one angle.

0:31But it is enough. And the reason it is enough is worth watching, because the same reason rescues a much harder job later on. Start where every construction starts. By drawing the thing badly. On purpose, badly. A quick freehand rectangle. Nothing measured, nothing straight. Label the corners A, B, C and D, going round. Draw the line from A across to C, and mark the sixty degrees where it leaves the bottom.

1:05This sketch is not an attempt at the answer. It is somewhere to put what you know. And looking at it, something is already obvious that was not obvious in words. The sixty degrees sits inside the corner at A. But the whole corner at A is a right angle. Ninety. So the piece left over, between the line and the left-hand side, has to be thirty. Nobody measured that. It had nowhere else to be.

1:36Now the real drawing. Draw the bottom side, from A to B. And here is the surprising part. Make it any length you like. Six centimetres. Ten. Whatever fits on the page. It genuinely does not matter, and that is not sloppiness. The instruction fixed an angle, and an angle fixes a shape without fixing a size. Every rectangle whose diagonal leans at sixty degrees looks exactly like every other one. Just bigger, or smaller.

2:10Then at B, at the far end, raise a perpendicular. Straight up, at a right angle to the side you just drew. The corner C has to be somewhere on that upright. Somewhere on it. But where? Go back to A and set off a ray at sixty degrees from the bottom side. Watch what happens as that ray travels. It is climbing, and the upright at B is standing still, and sooner or later the two of them have to meet.

2:42They meet once. Exactly once. Slide a point up and down that upright and the angle back at A changes at every step. A little higher and it is more than sixty. A little lower and it is less. Only one height on the whole line gives sixty degrees, and the ray walks straight to it. Mark it C. Two corners were given. A third has just been found. Three corners down, one to go. Where can D possibly be?

3:15D is joined to A, and the corner at A is a right angle. So the side running from A to D must set off at right angles to the bottom. Raise that perpendicular at A. D is on it. And notice what the drawing has just handed you for nothing. That thirty degrees, the leftover from the ninety at A, is now sitting there on the page, between the ray and the new line.

3:42It was never measured. It arrived because sixty and thirty make a corner. D is on that line. The only question left is how far up. Two ways to settle that, and both are worth doing. The first. Go to C, turn a right angle off the upright, and draw back across the page. Where that line crosses the perpendicular at A, that is D. That works because every corner of a rectangle is a right angle, C included.

4:15The second. Open a compass to the length from B up to C. Carry it over to A, and step that same length up the line. That is D. That works because opposite sides of a rectangle are equal, so the side from A to D has to match the side from B to C. Two different properties. Two different tools. Two different arguments. Mark both points, and they land on top of each other. Not close. The same point.

4:48Now the harder job, and the one all of this has been building towards. This time the instruction is not an angle. It is two lengths. But not two sides. One side, five centimetres. And the diagonal, seven centimetres. Straight away there is a temptation. Seven is the longest thing in the figure, so draw that first and build around it. You cannot. Think about what you would actually put on the paper.

5:18You know the diagonal is seven long. You do not know where either of its ends is. A length with no position is not something a pencil can draw. So begin with what can be placed. The side. Five centimetres, along the bottom. Call the left end D and the right end C. Two corners, fixed and finished. Those are not moving again. Now the corner above C. Call it B. What is known about B?

5:50Two things. It sits directly above C, because the corner at C is a right angle, so B is somewhere on the upright raised at C. And it is seven centimetres away from D, because that was the diagonal. Two conditions. Between them they ought to pin B down completely. They do. The difficulty is getting a pencil to it. Here is what almost everybody tries. Put the zero of the ruler on D and swing it. Let the far end drag up the upright at C, and watch the reading where the two cross.

6:29Six point four. Six point eight. Seven point one, too far, back down. And it does close in, because that reading only ever climbs as you slide upwards. It passes seven once, and never comes back to it. But try to land on seven exactly. Step up the line a millimetre at a time, all the way, eight hundred positions, and not one of them reads seven. The answer is not sitting at a whole number of millimetres.

7:02So the ruler gets close, then closer, and then stops. Do it again tomorrow and you would stop somewhere slightly different. That is not a construction. That is a hunt. The way out is to stop looking for the point. Ask a different question instead. Not, where is the corner that is seven from D. But, where are all the places that are seven from D? Put the compass point on D, open it to seven, and go the whole way round.

7:34Every single point on that circle is exactly seven centimetres from D. That is what a circle is. And it is easy to walk past, because a circle usually turns up as a shape to be drawn rather than as an answer to a question. Here it is an answer. It is the complete list of everywhere B is allowed to be. And the second condition already had a list of its own.

8:03The upright at C. Every point on that line, and nothing else. So B is on the circle, and B is on the line. There is exactly one place that manages both. They cross. And that crossing is the corner. Mark it B. Raise the perpendicular at D as well, draw across from B to meet it, and the figure closes. Now check it, because a marked point is not yet a finished job.

8:32Are the opposite sides equal? Measure them. Is every corner a right angle? Every one of them was drawn as one. And the side nobody ever specified comes out at four centimetres and nine millimetres. Just under five. It was never asked for and it was never chosen. The two given numbers had already decided it. One last thing, and it is the part that shows what the circle was really doing.

9:04Try the same job with a side of five and a diagonal of four. Draw the side. Raise the upright at C. Compass on D, open to four, and swing. The circle does not reach. It closes up short of the line, and the two of them never touch. There is no such rectangle. The circle said so immediately, without anyone trying and failing. A sliding ruler would have hunted for a long while before admitting that.

9:35The diagonal of a rectangle has to be longer than its sides, and that circle is exactly where you can see it. One last economy. Once you trust the crossing, stop drawing the whole circle. Draw the short arc, just where you expect it to cut. Same point. Far less ink.