Constructing a 90° angle at a chosen point on a line

0:00Here is a straight line, and here is a point sitting on it. Call it O. You are asked for a right angle at O. Not near O. At O, with one arm running along the line you already have. A set square would settle this in a second, and so would a protractor. You have neither. You have a compass, and a straight edge with nothing written on it.

0:24That turns out to be enough, and the reason it is enough is worth more than the drawing. Because you already know how to do this. You just do not know yet that you know it. Here is what you already know how to do. Given a segment with two ends, you can find the line that cuts it in half and meets it square. Open the compass past halfway, swing from one end, swing from the other, join the crossings.

0:54Now try to start that here. The very first instruction says open the compass past halfway. Halfway of what? There is no segment on this board. There is a point, and a line running off in both directions. So the recipe stalls on its first word, and that is not a small gap. That is the whole difficulty. The way out is not another recipe. It is reading the one you have from the other end.

1:25What does that construction actually do? It takes a segment, and it hands you back the middle. You have the middle. You want the line. So make a segment. Manufacture the two ends yourself, one on each side of O. Then the construction you already know will hand back exactly the thing you were asked for. That reversal is the entire topic. Everything after this is detail. Extend the line a little each way first, so there is room to work.

1:59Now open the compass. To anything at all. There is no correct opening here, and nothing on the board tells you what to pick. Whatever you choose, lock it, and do not touch it again. Put the point on O and mark the line to the left. That mark is X. Put the point on O again, same opening, and mark to the right. That mark is Y. Two marks, one opening used twice, and O is now exactly halfway between X and Y.

2:31Not because anybody measured. Because one distance was used twice, which is the only thing a compass is good at. Now look at what you have built, and ask the old question about it. Where is the line that cuts X to Y in half and meets it square? Every point that is equally far from X and from Y sits on that line. That was the whole of the last idea.

2:59So test O. How far is O from X? One opening. How far from Y? The same opening. O is equally far from both. O is on the line you are hunting. And you have not drawn a single arc yet. That is the dividend the reversal pays. You begin already owning a point of the answer. Which turns the rest of the job into counting. A straight edge needs two points. Give it one and it can still swing anywhere it likes through that point.

3:34Give it a second and there is exactly one line it can rule. You have one already. O. So the arcs are being asked for one more point. One. Not two. And half of the construction you memorised has just become unnecessary, for a reason you can say out loud. So here is the whole thing. Open the compass wider than the step you took, wider than O out to X.

4:02Swing an arc from X. Swing an arc from Y with that same opening. They cross above the line, at a point. Call it A. Lay the straight edge on O and on A, and rule. That is your right angle, and you are finished. Notice what is missing. There is no crossing below the line, and there is no need for one. Draw it out of habit if you like. It will land exactly where you expect, and it will tell you nothing you did not already have.

4:35One thing to be careful about, because it is the easiest thing to get backwards. The right angle is at O. It is not at A. The corner at A is nowhere near square, and nothing in the construction ever claimed it was. A has one job. To be a second point, so the straight edge knows which way to go. Open the compass wider and A climbs higher up.

5:01Open it narrower and A drops. The line through O is the same line every time. A moves. The answer does not. The angle you were asked for is at the point you were handed. Put the two constructions beside each other and the missing arcs explain themselves. The first one began knowing nothing. Not one point of the answer was in hand. So it had to manufacture two of them, one crossing above and one below, and join them.

5:33This one began holding a point already. So that is what the second pair of arcs was ever doing. Not squareness. Not accuracy. It was buying a second point, because two points are what a straight edge costs. Get one for free and you only ever needed the one pair. Now go back to the first move and loosen it, because it looks fussier than it is. Does the step from O out to X have to be some particular length?

6:05Step out five, then cross with an opening of thirteen. The crossing lands twelve above the line. Step out twelve instead, cross with thirteen again, and the crossing drops to five above. Step out fifteen, cross with seventeen, and it lands eight above. Three different steps. Three crossings in three different places. And all three of them rule the very same line through O. So the size never mattered. What mattered is that the two steps were equal, because that is what put O in the middle.

6:42A student reaching for a ruler here has misread the question. One more version, and it is the same move a third time. Now the point is not on the line. It is P, floating somewhere above it, and you want the perpendicular from P down to the line. Put the compass point on P and open it until the arc will reach past the line. Swing. It cuts the line in two places.

7:11Both of those cuts are on one arc from P, so P is exactly as far from one as from the other. Which means P is equally far from two points of the line. And you know where such points live. P is on the perpendicular bisector of the piece between the cuts. So bisect that piece, the way you already can, and the line you draw runs through P and meets the line square.

7:39Handed a point, you manufactured a segment it sits equally far from. Same reversal, third time. It is worth noticing that none of this actually needs a compass. Strip the construction down to what it uses, and it uses one thing. Hold a distance, and turn. A rope with a peg does that. Peg it at O, pull it tight, scratch the ground to the left and to the right. There are X and Y.

8:09Take a longer rope. Peg at X and scratch an arc. Peg at Y and scratch another. Where they meet is A. Pull a line from O through A and you have a right angle in the ground. Nothing was read. Nothing was numbered. A rope with a knot in it is a compass that happens to be made of rope. So why build a right angle this way, when a set square would have done it?

8:37Because of what happens next. A right angle is a hundred and eighty halved. The next thing anybody wants is half of that again. Forty five degrees. And you get it by cutting this angle in two. The same word as before, bisecting, doing the same work one dimension up. But you cannot cut an angle in half until you have one to cut. That is what this construction really is. Not a trick for right angles. The seed.

9:08And it was built from a point you were given, a distance you chose at random, and one crossing.