Why a rectangle's diagonals are equal, and when they split the corners evenly

0:00Here is a rectangle. Its corners, read in order round the outside, are P, Q, R and S. Join P to R. Then join Q to S. Each of those lines connects a pair of opposite corners, and together they have a name. They are the diagonals. A rectangle has exactly two, and drawing both takes about three seconds. What happens next is that the figure becomes considerably more interesting than it was.

0:31Because those two lines have cut every one of the four corners into two pieces. That is eight new angles to think about. But there is a simpler question to deal with first. The simpler question is whether the two diagonals are the same length as each other. Before measuring anything, commit to an answer. Most people say yes, and most people are right. Lift the two lines off the figure and lay them side by side.

1:03The same length. Exactly the same, not nearly. But notice carefully what has just happened and what has not. You have checked it on one rectangle. Hold on to that. We will come back at the end to what it would take to be sure of it everywhere. Now those eight angles. Give them letters, so that they can be talked about one at a time. Up at P, call the angle against the top side d, and the one against the left side c.

1:35At Q, e against the top and f against the right. Down at S, b against the left side and a against the bottom. And at R, g against the right and h along the bottom. Eight letters. It looks like eight separate measurements waiting to be taken. It is worth guessing now how many different numbers are really going to turn up. Here is the question that matters most, and it deserves an answer before any measuring at all.

2:07Look at the corner at P. The diagonal has cut it into c and d. Are those two pieces equal to each other? Almost everybody says yes. The line runs from one corner to another, it looks for all the world as though it goes down the middle, and the picture is extremely persuasive. Ask the same thing about the corner at R, where the pieces are g and h.

2:34Keep your answer somewhere. It is the most common wrong belief on this whole topic, and it is worth watching it fail. So measure. Take a rectangle seven centimetres across and four centimetres high. Start at P. The angle d, the one leaning on the long top side, comes out at about thirty degrees. And c, leaning on the short left side, about sixty. Move round to Q. e is thirty. f is sixty.

3:06Down at S: a is thirty, and b is sixty. At R: h is thirty, and g is sixty. Eight measurements taken, and written down. Now stop and look at what is written down. There are not eight numbers there. There are two. Thirty turns up four times, and sixty turns up four times. And the four thirties are not scattered about at random. Every single one of them leans on a long side of the rectangle.

3:38Every one of the sixties leans on a short side. Colour them in and the whole figure sorts itself into two families of four. Eight angles had sounded like eight numbers to go and find. That is the trap, and the way out of it was to look at all the readings together instead of one at a time. Eight things to measure turned out to be one thing to measure, and a great deal of copying.

4:07Take any single corner and look at its two pieces. Thirty and sixty. They add to ninety. And that is not something the measuring discovered. It was never in any doubt. The two pieces are the corner, and every corner of a rectangle is a right angle. So the moment you know one of the two numbers, you know the other one by subtracting. And because of all that copying, knowing one of them tells you all eight.

4:37One measurement. That is the entire content of those eight readings. Which settles the question from before. Is c equal to d? No. Thirty is not sixty. Is g equal to h? Also no. The diagonal does not cut the corner in half. It cuts it into a large piece and a small piece. And once you know to look for it, you can see which way it leans. The diagonal hugs the long side and stands well clear of the short one.

5:12The persuasive picture was wrong, and it was wrong by thirty degrees. But watch what happens if you start changing the rectangle. Keep the height exactly as it is, and drag the width. Make it very wide and very flat. The angle on the long side shrinks away towards nothing, and the one on the short side climbs towards ninety. Now squeeze it the other way, tall and narrow, and the two of them swap roles completely.

5:45Which means that somewhere in between, they have to pass each other. And they do. Exactly once. Notice that the two of them never move the same way. When one grows the other shrinks, because between the two of them they always have to make ninety. And two numbers moving in opposite directions can only ever meet once. Not twice. Not over a stretch of widths. There is precisely one width at which the two values agree, and on either side of it one of them is always the larger.

6:22At that one width, both of them read forty-five. Forty-five and forty-five, adding to ninety, exactly as they have to. Now look at the figure itself at that instant. Its width has become equal to its height. It is a square. So a diagonal that halves its corners is not a fact about rectangles at all. It is a test. Ask for a rectangle whose diagonal splits the corner into forty-five and forty-five, and you have asked for a square, whether you meant to or not.

6:57Ask for fifty and forty instead and you get an ordinary rectangle, a little taller than it is wide, and nothing special happens. One more thing changes at that same instant, and it is easy to walk past. On an ordinary rectangle, the two diagonals do not cross each other at a right angle. On the square, they do. There is one more thing worth pulling out, and it is about how the measuring was done rather than what it found.

7:31Before a single number was collected, a decision was made about what to write down. The two side lengths, and the eight angles. Ten columns, named in advance. That sounds like bookkeeping. It is not. If the side lengths had been left out, everything found in the last few minutes would have been invisible. Because the pattern is about which side each angle leans on, and without the sides there is nothing to lean on.

8:01Choosing what to record is part of the experiment, and it gets done before there are any results around to be tempted by. Which brings back the thing left hanging at the start. The two diagonals came out the same length. On one rectangle. Suppose you measure six more, and all six agree. Are you now certain it holds for every rectangle that anyone will ever draw? No. Six is not all of them. Neither is six hundred.

8:34But look at the figure one more time. Each diagonal is the longest side of a right-angled triangle, and the other two sides of that triangle are a width and a height of this rectangle. The same width, and the same height, both times. That is not a measurement. That is a reason. And it covers every rectangle at once, including all the ones nobody is ever going to draw.