Constructing the perpendicular bisector of a line segment

Right now your phone is talking to one mast. Walk far enough along the road and it quietly hands you over to a different one. You never feel it happen. But there is a line out there where the balance tips. On one side, the first mast is nearer. On the other side, the second one is. And standing on the line itself, you are exactly the same distance from both. That line is a real, drawable thing, and drawing it exactly is what this video is for.

Before any method, the picture you are aiming at.

Here is a line segment, with its two ends labelled A and B. And here is the finished job. A straight line crossing it, cutting it exactly in half, at exactly ninety degrees.

That line has a name. It is the perpendicular bisector of A B. Perpendicular means at right angles. Bisector means it cuts something into two equal halves. Put those together and you have a line that does both at once.

Now look at what else is still on the page. Four curved arcs, two above the segment and two below. They are not decoration, and they are not mess to tidy away. They are how the line was found.

A quick one for you. Three different lines are drawn across the segment A B. Line one crosses exactly halfway, but leans over at about seventy degrees. Line two crosses at right angles, but a third of the way along, nearer to A. Line three crosses at right angles, exactly halfway. Which of the three is the perpendicular bisector?

Have a think. I'll wait.

The answer is line three. It has to be both things at the same time. Line one is halfway, but not square. Line two is square, but not halfway. Only line three is halfway and square.

Halfway, and square. Both, or it is not a perpendicular bisector.

Now, why a pair of compasses can find that line when a ruler and a good eye cannot.

Open your compasses and fix the width. Put the point on A, and sweep. Every single dot on that arc is exactly the same distance from A. That is all a pair of compasses does. It holds one distance, and it refuses to let go of it. Now do not touch the width. Move the point over to B and sweep again. Every dot on this second arc is that same distance from B. So think about where the two arcs cross. That crossing point is sitting on both arcs at once. So it is that distance from A, and the very same distance from B. It is equidistant from the two ends. That word just means the same distance from both. Do that above the segment and below it, and you get two points that are each equally far from A and from B. Draw the line through them, and every point along it is the same distance from A as from B. It has no choice but to cross A B at the midpoint, and to cross it square. And that is the mast line from the start. The set of places where two things are exactly equally far away.

Which gives you three words to carry into the exam hall. Same width, both ends. Change the width between the two sweeps and the crossing is no longer fair to both ends, so the line you draw is wrong.

Now the method, one step at a time.

Step one. Open your compasses to more than half the length of A B. You do not need to measure that. Just open them so that, with the point on A, the pencil reaches past the middle. About three quarters of the way along is a safe choice.

More than half is the bit that matters. Open them too narrow and the two sets of arcs never reach each other, so nothing crosses, and there is nothing to draw through.

Step two. Point on A. Sweep an arc above the segment, and an arc below it. Step three. Do not touch the screw. Lift the point over to B and sweep again, above and below. Now the arcs cross in two places. One above the segment, one below. Call them P and Q. Step four, and this is the step that gets left out. Lay your ruler on P and Q, and draw the straight line right through both of them. Run it a little past each crossing. The arcs on their own are not the answer. The line is the answer. And one instruction to hold on to at the end. Leave the arcs alone. Do not rub them out to make the page look tidy.

So the chant is finished. Same width, both ends, join the crosses.

Time to put that on a real exam question.

Two phone masts, C and D, are shown nine centimetres apart on a map. Using only a ruler and compasses, construct the perpendicular bisector of the line C D, showing all your construction arcs.

This one is yours first. Grab a ruler and a pair of compasses, draw C and D nine centimetres apart, and construct it before I do.

Pause here and have a go. I'll wait.

Here it is. Nine centimetres apart, so half is four and a half. Open the compasses to about seven, comfortably more than half. Point on C, arc above, arc below. Same width, point on D, arc above, arc below. Ruler through the two crossings, and the line drawn right through. Arcs left on the page.

And here is how you check yourself, with nobody marking it. Measure from C to where your line crosses. Four and a half centimetres. Measure from there on to D. Four and a half centimetres. Put a protractor on the crossing. Ninety degrees.

Now the exam side of this one, because it is marked in two halves. The arcs earn the method mark. They are the evidence that you used the construction. The line earns the accuracy mark. Two separate things, two separate marks. Which is why a line drawn by eye is an expensive habit. It can sit almost perfectly on the page and still leave the method mark unearned, because nothing on the paper shows how it got there. Examiners report it year after year.

Over half of candidates answered this question, though generally only the high performing candidates used compasses to construct the perpendicular bisector of AB.

Side by side, then. The eyeballed line, no arcs, at most one mark. The constructed line, arcs showing, both marks. Almost the same line on the paper. A different score.

And it goes wrong the other way round too. The arcs go on beautifully, and then the line never gets drawn.

Some drew the correct arcs, but then didn't draw a perpendicular bisector (these were given B1).

B one there means one mark out of the two. All that compass work, and the five seconds with a ruler is what was missing.

One more line from the reports, and it is the reason this video opened with the finished picture instead of the method.

Many candidates seemed unfamiliar with this type of construction and very few candidates provided a response consistent with a perpendicular bisector of AB.

So know the shape before you need it. Four arcs, two crossings, one line through them.

Right. The whole construction on one page. A perpendicular bisector cuts a line segment exactly in half and crosses it at ninety degrees. Halfway and square. It works because a pair of compasses holds one distance. The same width swept from both ends puts each crossing exactly as far from one end as from the other. The line through those crossings is every point that is fair to both. Same width, both ends, join the crosses. And the arcs stay on the paper, because they are your working. If you want to prove it to yourself tonight, draw any line segment at all, construct its perpendicular bisector, then measure. The ruler should say the two halves match. The protractor should say ninety degrees.

Next in the chapter: constructing the bisector of an angle. The same compass thinking, turned on a corner instead of a straight line.

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