KS3 · Maths

Perpendicular bisector of a line segment

No ruler markings, no protractor — yet your line has to hit the exact middle at exactly 90°. Sounds impossible? A pair of compasses and one clever shape do it every time.

Maths · Constructions

Pick a compass width — any width
9.0 cm9.0 cm9.0 cm9.0 cm6.0 cm6.0 cm90°ABMPQ

Width from A to P 9.0 cm. Width from B to P 9.0 cm. Width from A to Q 9.0 cm. Width from B to Q 9.0 cm. AM 6.0 cm. MB 6.0 cm. Angle at M 90°. Relationship: Every point on line PQ is the same distance from A as from B — so that's where equal arcs cross. And line PQ cuts AB into two equal halves at 90°.

Width from A to P9.0 cmWidth from B to P9.0 cmWidth from A to Q9.0 cmWidth from B to Q9.0 cmAM6.0 cmMB6.0 cmAngle at M90°Drag P or Q

Every point on line PQ is the same distance from A as from B — so that's where equal arcs cross. And line PQ cuts AB into two equal halves at 90°.

AB is 12 cm. P is where two arcs of the same width — one from A, one from B — cross above AB, and Q is where two equal arcs cross below it. The faint lines show each compass width. They start at 9 cm, a sensible choice. Drag P or Q to try others: 7 cm works, but the crossing points crowd close to AB — and 20 cm wouldn't even fit on the page.

Watch out: Drag P right down onto AB and the widths read 6.0 cm — exactly half of AB. That's as small as they can go: the arcs only touch, on the segment itself. Any narrower, like 3 cm, and the arcs from A and B never meet at all.

Why it works

?

Reason it through

Why does the line through the two crossing points cut AB in half at exactly 90°?

Link 1 of 4

First link · your turn

Both sets of arcs use the same compass width. So what's true about each crossing point?

2
Locked — reveal the link above first
3
Locked — reveal the link above first
4
Locked — reveal the link above first

The quick way: arcs only

Problem

Construct the perpendicular bisector of a line segment AB that is 5.5 cm long, using arcs instead of full circles.

Spot the mistake

Where does Sam's construction go wrong?

Construct the perpendicular bisector of a line segment XY that is 10 cm long, using full circles.

Sam's method — which line goes wrong?

What do you think?

Why bother with compasses?

Jo has to draw the perpendicular bisector of a 7 cm line. She reckons there's an easier way than arcs.

Which is closest to what you think right now?
How sure are you?

Predict, then check

Construct one side's bisector at a time, ignoring the other two sides while you work.

You construct the perpendicular bisector of each side of a triangle. What do the three lines do?

WHAT YOU'VE LEARNED

A quick recap of today's lesson.

Two equal arcs, one straight line — and the exact middle and the right angle come guaranteed.

What you need to know

  • What 'bisect' and 'perpendicular' mean, and what a perpendicular bisector does to a line segment.
  • How to construct a perpendicular bisector with a ruler and compasses, choosing a width that works.
  • Why the construction works — the rhombus hiding inside it — and how to use it to find a triangle's circumcentre.

The big picture

The perpendicular bisector of a line segment cuts it into two equal halves at 90°. To construct it, draw arcs of the same width — more than half the segment — from both endpoints, then draw a line through the two points where they cross. It works because those points and the endpoints make a rhombus, whose diagonals always cross at right angles and cut each other in half.

Key points

1To bisect means to cut into two equal parts; perpendicular means meeting at right angles.
2A perpendicular bisector cuts a line segment into two equal halves with a line at 90° to it.
3Set the compasses to more than half the length of the segment, and keep the same width for both endpoints.
4Put the needle exactly on each endpoint, and draw the line through BOTH crossing points.
5It works because the crossing points and the endpoints form a rhombus, and a rhombus's diagonals cross at 90° and cut each other in half.
6A construction uses no measuring — keep every arc to show how it was made.

Worked example

Problem

A line segment is 9.4 cm long. How long will each half be after you bisect it? And which of these compass widths would give you a perpendicular bisector: 3 cm, 4.7 cm or 6 cm?

⚠ Watch out

Putting the compass needle on the line but not exactly on an endpoint. The arcs still cross and the line still meets the segment at 90°, but it misses the midpoint — so you've drawn a perpendicular, not the perpendicular bisector.

🧠

Memory hook

Miss, touch, cross. Too narrow and the arcs miss; exactly half and they touch; wider than half and they cross twice — and the line through both crossings is your answer.

✓

Check yourself

A line segment is 8 cm long. Would compasses set to 4 cm let you construct its perpendicular bisector? Say what would happen, and what you would change.

Flashcards

(12)
What does 'bisect' mean?
To cut or divide something into two equal parts.
What does 'perpendicular' mean?
Meeting at right angles (90°).
What does the perpendicular bisector of a line segment do?
It cuts the segment into two equal halves, with a line at 90° to it.
What is a rhombus?
A parallelogram with all four sides the same length.
What is special about the diagonals of a rhombus?
They cross at right angles and cut each other into two equal halves.
What is a construction?
A way of making shapes and angles without measuring — just a ruler for straight lines and a pair of compasses.
How wide must the compasses be to bisect a line segment?
More than half the length of the segment — and the same width from both ends.
Not sure what width to use?
Set the compasses to the whole length of the segment. That is always more than half.
What is true about each point where the two sets of arcs cross?
It is the same distance from both endpoints of the segment.
Why must the line go through BOTH crossing points?
One point alone gives no guarantee that the line is at 90° or that it passes through the midpoint.
Why keep the construction arcs?
They show the line was constructed, and how accurately. A line with no arcs doesn't show a construction.
What is the circumcentre of a triangle?
The point where the perpendicular bisectors of all three sides meet — the centre of a circle through all three corners.

Tap any card to flip it, or use Study as deck to go through them one at a time. In the full lesson these run as a spaced-repetition deck — you rate each card Hard, Good or Easy and the tricky ones keep coming back until they stick.

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How this lesson was checked. This KS3 Mathslesson was published through Lightbulb Learning's human-designed editorial process — the educational standards, accuracy rules and publication checks it must pass were authored and approved by Philip Halpin. It passed subject-specific assessment, automated educational checks and technical publication verification before going live (publication checks completed 2 October 2026). Published pages are monitored, human spot-checking is ongoing across the lesson library, and anything found wrong is corrected or withdrawn. How our lessons are made and checked. Spotted a mistake? Email hello@lightbulblearning.co and we'll review it.