Draw the perpendicular bisector

The idea

With an unmarked ruler and compass, draw equal-radius arcs from X and Y that meet above XY, and again below it; the line through the two meeting points is the perpendicular bisector, which also locates the midpoint.

Find the middle without measuring

A line segment XY is drawn on paper. You want its exact middle. You have only an unmarked ruler and a compass.

How could you find the middle?

• Guess, and then check with a marked scale
• Draw equal arcs from X and from Y, and join the points where they meet
• It cannot be done without a marked scale

Build it yourself

Do each step. The compass draws an arc only when its opening is right for that step. Remember: the arcs from X and from Y need the same opening.

Two points, one line

The arcs above XY cross at A. The arcs below cross at B. Each of these points is the same distance from X and from Y. So A and B both lie on the perpendicular bisector. Two points fix a line, so the ruler through them draws it.

The line also crosses XY at its midpoint. This is more exact than measuring with a marked scale.

Take an opening more than half of XY. Draw arcs of the same radius from X and from Y, above XY and again below it. The line through the two meeting points is the perpendicular bisector of XY. It also finds the midpoint.

• Step | What it gives
• Equal arcs from X and Y, above XY | A, equally far from X and Y
• Equal arcs from X and Y, below XY | B, equally far from X and Y
• Ruler through A and B | the perpendicular bisector

Notes

Take an opening more than half of XY. Draw arcs of the same radius from X and from Y, above XY and again below it. The line through the two meeting points is the perpendicular bisector of XY.

Check yourself

A segment XY is 8 cm long. What is the smallest whole number of cm to which the compass can be opened so that the arcs from X and from Y meet?

Answer: 5

Half of 8 cm is 4 cm. The opening must be more than 4 cm, so the smallest whole number is 5 cm.

Can both pairs of arcs be drawn on the same side of XY, for example both above it (one pair with a bigger opening)?

Ravi opens the compass to 4 cm for the arc from X, but to 5 cm for the arc from Y. The arcs cross at P. Is P on the perpendicular bisector of XY?

XY is 9 cm long. AB crosses XY at O. How many cm is it from X to O?

Answer: 4.5

AB passes through the midpoint of XY, so XO is half of 9 cm: 4.5 cm.

• No. One point must be above XY and one below it.. Not necessary. Two points above XY, each equally far from X and Y, also fix the line.
• Yes. Each pair of crossing arcs gives a point equally far from X and Y. Two such points are enough to draw the line. — correct. Yes! Any two points that are the same distance from X and from Y lie on the perpendicular bisector.
• Yes, but the line must be drawn with a protractor.. No protractor is needed. Any two such points fix the perpendicular bisector.
• Yes. Any point where two arcs cross is on the bisector.. Only if the arcs have the same radius. Here P is nearer to X than to Y.
• No. P is 4 cm from X and 5 cm from Y, so it is not equally far from them. — correct. Yes, that is why! The two arcs of one pair must have the same radius.
• Yes, if XY is long enough.. The length of XY does not change this. P is 4 cm from X and 5 cm from Y.