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Chapter 6 · Constructions and Tilings

The perpendicular bisector, and the equidistance property that justifies it

यह वीडियो हिंदी में भी · Watch in Hindi

Bisecting, and building a right angle9 min

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9 min.

Also recorded in Hindi.Englishहिन्दी

A compass cannot be told what perpendicular means. All it can do is hold one distance while it turns.

The idea

A compass cannot be told what perpendicular means. All it can do is hold one distance fixed — so the only thing those four arcs actually manufacture is points that sit as far from X as they do from Y. Everything else is argued. Congruent triangles are what convert "equally far from both ends" into "cuts XY in half" and "meets XY square on", and once you have followed that argument the construction stops being a ritual you copy: you can see why the radius above need not match the radius below, why one crossing point on each side is enough, and why a compass beats a marked scale at the very job the scale looks built for.

What you should be able to do

  • State what bisection means and what the extra word perpendicular adds
  • Explain why the four arcs of the construction produce points that are the same distance from both ends of the segment
  • Reconstruct the two-step congruence argument and say which condition each step uses
  • Explain why two equal angles that together fill a straight angle must each be 90°
  • Justify the converse: a point equally far from both endpoints has to sit on their perpendicular bisector
  • Carry out the construction with a compass and an unmarked ruler
  • Decide, with a reason, whether the arcs above and below need the same radius
  • Say why this construction locates a midpoint more reliably than measuring does

Words to know

TermDefinition in one lineFirst introduced
bisectioncutting something into two identical partsprinted in bold in §6.1, Part II, p.137, and restated in the SUMMARY, p.163
perpendicular bisectorthe line that cuts a segment in half and meets it at a right angleprinted in §6.1, Part II, p.137, and throughout pp.139–142
midpointthe point that divides a segment into two equal halvesprinted in §6.1, Part II, pp.137, 141, 142
arcpart of a circle drawn from a centre at a fixed radiusprinted in §6.1, Part II, pp.136–137, 139–140
radius / radiithe fixed compass opening an arc is drawn withprinted in §6.1, Part II, pp.136–137, 139–140
compassthe instrument that holds one distance while it turnsprinted in §6.1, Part II, pp.139–141
unmarked rulera straight edge carrying no scaleprinted in §6.1, Part II, pp.139–140
congruentsaid of two figures that match part for partprinted in §6.1, Part II, pp.137–138
SAS congruence conditiontwo sides and the angle between them, forcing congruenceprinted in §6.1, Part II, p.138
straight anglethe 180° angle made by two opposite rays at a pointprinted in §6.1, Part II, p.137
supporting linea line drawn only to place a figure, and not part of itprinted in §6.1, Part II, pp.136, 140
line of symmetrythe line a figure can be folded along to match itselfprinted in §6.1, Part II, p.136
equidistantequally far from two given pointsthe explanation's shorthand, not printed in this chapter — the chapter always says it the long way instead (pp.136, 139, 140)

Where people slip up

  • "The arcs above and below must be drawn with the same radius." This is the first thing the chapter's own Figure it Out attacks (p.140, question 1). They need not. Any point above that is equally far from both endpoints will do, and any point below likewise; two such points fix the line.
  • "You need all four arcs — that is what makes it work." Four arcs give two points, and two points are what a line needs. The chapter makes the same observation do real work on p.141, where one of the two points is already known and the second pair of arcs is dropped.
  • "Both crossings have to be on opposite sides of XY." Question 2 on p.140 asks exactly this and invites a construction rather than a rule. Nothing in the argument mentions sides; it only mentions distances.
  • "It is perpendicular because it looks perpendicular." The whole of pp.137–138 exists because looking is not enough. The right angle is deduced from two equal angles that fill a straight angle, not measured off the drawing.
  • "Bisecting is just finding the middle by measuring." The chapter says the compass method is the more accurate of the two (p.141). Measuring introduces a reading error twice over; the compass never reads anything.
  • "Bisection is a thing you do to line segments." The definition on p.137 is wider than that, and the chapter cashes it in two pages later on angles. Keep the definition general when you state it.

Worked answers to this chapter’s exercises · this video explains Figure it Out · 1 Q1, Figure it Out · 1 Q2, Figure it Out · 1 Q3, Figure it Out · 1 Q4

Transcript1,353 words

Here is a shape you have almost certainly drawn: two arcs meeting at two sharp corners, like an eye. To draw it you put the compass point somewhere above the line, swing an arc, then somewhere below, and swing another. And the question nobody asks out loud is where exactly those two somewheres go. Most people put them by eye. Nudge, look, nudge again, until the two halves look the same.

That is a guess, and it works often enough that you stop noticing it is a guess. So let us stop guessing, and find out what those two points actually have to be. Call the two corners X and Y, and call the centre of the upper arc A. Here is the thing about a compass: while you swing it, the distance does not change. So the arc from A passes through X and through Y, which means A is exactly as far from X as it is from Y.

Not roughly. Exactly, because it is the same compass opening both times. Now the lower centre, B. Same story: B is as far from X as it is from Y. And if you want the two arcs to match each other, you used the same opening for both. So four lengths, all equal: A to X, A to Y, B to X, B to Y. That chain is the whole of what the compass gave you. Everything else has to be argued.

Watch it happen with nothing decided in advance. Pick an opening, wider than half of X to Y, and swing an arc centred on X. Keep the same opening. Swing an arc centred on Y. They cross in two places, one above the line and one below. Mark the upper crossing A and the lower one B. Notice what you never did. You never measured an angle, and you never found a middle.

You held one distance and turned. That is the only instruction a compass can take. Now take a straight edge and join A to B. Two things happen that you did not ask for. The join crosses X to Y exactly at its middle. And it crosses square on, at a right angle. Neither of those was in the instructions. You asked for one distance, held four times. Which is exactly why they need explaining rather than admiring.

Cutting something into two identical parts has a name. Bisecting. The line A to B bisects the segment X to Y. And that word is wider than it looks. You can bisect an angle just as well as a length. The extra word, perpendicular, is doing the second job: it says the cut arrives at a right angle. Put them together and you have the perpendicular bisector of X to Y.

Two claims in one name, and so far you have seen both and proved neither. So here is the doubt worth having. Was that a fluke of this particular drawing? Make X to Y three times longer. Open the compass much wider. Do it again. It still works, and a hundred more drawings would still not settle it. A drawing shows you that it happened. It cannot show you that it must.

For that you need an argument, and the argument is shorter than you would think. Look at the two triangles A B X and A B Y. The side A to B is not two equal sides. It is the same side, sitting in both triangles at once. A to X equals A to Y. That was the first link in the chain. B to X equals B to Y. That was the second.

So all three sides of one triangle match all three sides of the other. Three matching sides is enough. There is no second triangle you could build from three given lengths. The triangles are identical, and every part of one equals its partner in the other. The part we want is the angle at A, which has just been cut into two equal halves. Call the point where A B crosses X Y the letter O.

Now the two smaller triangles: A O X and A O Y. A to X equals A to Y, still. A to O is shared, sitting in both. And the angle at A between those two sides is the angle we just split in half. Two sides, and the angle between them. Between them is the part that matters. Give me those three things and the triangle has no freedom left. It can only close one way.

So these two triangles are identical as well. Which hands us O to X equals O to Y. The middle was not placed. It was concluded. One part left, and it is the one people take on trust. The two triangles being identical also makes the two angles at O equal. Now look at what those two angles sit on. X, O and Y are three points on one straight line, with O between the other two.

So the two angles at O fill a straight angle between them, and a straight angle is a hundred and eighty degrees. Two equal things filling a hundred and eighty. Each one is ninety. That is a right angle, and notice where it came from. Not from a protractor, and not from the drawing looking right. From a division. Now turn the whole thing around, because the useful version runs the other way.

Forget the construction. Take any point at all and ask one question: is it equally far from X and from Y? Here is a point that is. And another. And another. Keep going, and mark every point that passes the test. They do not scatter. They land on one line, and it is the line you already built. Which gives you something you can use as a tool. If you ever learn that a point is equally far from two others, you know where it lives.

And that is the sentence to keep, because the drawing is just one way of finding those points. So the whole construction, in three steps, with two tools on the table. A compass, which holds a distance. And a straight edge with no marks on it at all. One. Open the compass past halfway and draw arcs from X and from Y, above the line. Two. Same again below the line. You now have a crossing on each side.

Three. Join the two crossings. That line is your answer. The straight edge has no numbers because it does not need any. Nothing here is ever read. And now that you have the argument, look at how much of the recipe was never required. Do the arcs above and below need the same opening? Go back to the argument and look for the step that used it. There isn't one.

Thirteen above, fifteen below. Different openings, and the line comes out the same. Do the two crossings have to be on opposite sides? The argument never mentions sides either. It only ever mentions distances. Two crossings, both above the line, and again the same line. Four arcs feel like the minimum. What you actually need is two points, each equally far from both ends. One question is still fair. Why not just measure it and halve it?

Because a scale has to be read, and read twice, and every reading rounds to the nearest mark. Round one end up and the other end up as well, and the errors do not cancel. They survive the halving. With millimetre marks, that leaves your middle up to half a millimetre off, and you will never know which way. The compass reads nothing. It transfers a distance without ever naming it.

That is why it wins at the one job the ruler looks built for. So finish with something worth drawing. Four leaves meeting at a point, each one bounded by two arcs. Compass and straight edge only, and every centre in it is a point equally far from two others. One distance, held. Everything else, argued.

Where this fits

Taken from the notes each video was made from, not from the reading order — these are the ideas this one rests on and the ones that later rest on it.

Builds on

Comes up again in

Either side of this one

The book

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