KS3 · Maths

Constructing perpendicular through a point

Puzzle: draw a line through a dot at exactly 90 degrees to another line, with no protractor. The trick is to turn the dot into the middle of something.

Try it first

Where do two equal arcs meet?
2.0 cm2.0 cm4.6 cm3.0 cm63°ABPXYS

PX 2.0 cm. PY 2.0 cm. Arc width from X 4.6 cm. Arc width from Y 3.0 cm. Angle at P 63°. Relationship: Equal arcs from X and Y meet on a line through P at 90 degrees only when P is exactly halfway between X and Y.

PX2.0 cmPY2.0 cmArc width from X4.6 cmArc width from Y3.0 cmAngle at P63°Try 1: slide S until the two arc widths match. Try 2: drag Y, then match them again.

Equal arcs from X and Y meet on a line through P at 90 degrees only when P is exactly halfway between X and Y.

P is a point on line AB. X and Y are the two places where a circle centred on P crosses the line. S is where an arc from X and an arc from Y would meet, and the lines XS and YS show the compass width from each end.

Watch out: If the circle is so big that it doesn't cross the segment twice, P is no longer the midpoint of the two marks you end up with, and the line you build misses P. Draw a smaller circle, or extend the segment along its own direction so the circle can cross it twice.

Why it works

?

Reason it through

Why does the line through the two arc crossings go through P at exactly 90 degrees?

Link 1 of 4

First link · your turn

You draw a circle centred on P. Think of a pizza: the crust is the same distance from the middle all the way round. So what can you say about X and Y, and about P?

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

Do it on paper

Problem

Line AB has a point P somewhere along it. Construct the perpendicular to AB through P, using compasses and a ruler.

Your turn

Now P is off the line

Segment AB lies flat, and P is a point above it, just beyond end B. Construct the perpendicular to AB through P.

  1. P is above the line but past the end of the segment, so straight down from P we would land beyond B. We still want the perpendicular to AB through P.
  2. missing step
Which line is step 2?

Spot the slip

Where does this construction go wrong?

A student constructs the perpendicular to AB through a point P on AB. Their circle has a radius of 3 cm. Find the line where their method goes wrong.

A student's method — which line goes wrong?

Predict first

Rule so far: the perpendicular distance from a point to a line is the shortest distance to that line.

Picture segment AB lying flat, 6 cm long, with A on the left and B on the right. Point Q is 3 cm above the line and 2 cm to the right of B, so the perpendicular from Q lands 2 cm past B, where the segment has ended. Which is the shortest distance from Q to the segment AB?

WHAT YOU'VE LEARNED

A quick recap of today's lesson.

Make the point the middle of something, and a method you already know does the rest

What you need to know

  • What a perpendicular is, and why it doesn't have to go through the middle of a line
  • How a circle centred on your point turns the job into one you already know how to do
  • The order of compass moves, and the slips that spoil the construction
  • What 'shortest distance from a point to a line' really means, and the catch with a segment

The big picture

A perpendicular can cross a line at any point, so to build one through a point P you first draw a circle centred on P. The circle's two crossings are the same distance from P, which makes P the exact middle of a shorter segment, and you then bisect that segment as you did before. The perpendicular distance from a point to a line is the shortest distance to the line, with one catch when the line is only a segment.

Key points

1A perpendicular meets a line at 90 degrees, and it can cross at any point on the line. Only a perpendicular bisector has to cross at the midpoint.
2To construct a perpendicular through a point P on a segment, draw a circle centred on P, then bisect the shorter segment XY that the circle cuts off.
3For the four arcs, use one compass width, more than half the circle's diameter, from both X and Y.
4A point off the line, or close to an end of the segment, uses the same method. Make the circle big enough to cross the line twice, and extend the segment along its own direction if you need to.
5The perpendicular distance from a point to a line is the shortest distance to that line. For a segment it is the shortest distance only if the perpendicular lands on the segment. Otherwise the shortest distance is to the nearer endpoint.

Worked example

Problem

P is 4 cm from end A of a 10 cm segment AB, so it is 6 cm from end B. You will draw a circle centred on P. Choose a radius for the circle and a compass width for the arcs so the construction works without the circle running off the segment.

⚠ Watch out

Thinking a perpendicular must pass through the middle of the line, or that any circle will do. It can cross anywhere. The circle just has to cross the line twice (extend the line if needed), so your point becomes the midpoint of the shorter segment.

🧠

Memory hook

Circle on the point. Four equal arcs. One ruler line.

✓

Check yourself

P is 1 cm from end B of segment AB, and a circle of radius 3 cm on P would cross the segment only once. What do you do?

Flashcards

(13)
What does perpendicular mean?
Meeting at a right angle, 90 degrees.
Must a perpendicular to AB cross AB at its midpoint?
No. It can cross at any point on AB. Only a perpendicular bisector has to cross at the midpoint.
First move when constructing a perpendicular through a point P on a segment?
Needle on P, draw a full circle that crosses the segment on both sides of P.
Why draw the circle centred on P?
The two crossings are the same distance from P, so P becomes the midpoint of the shorter segment between them.
What compass width do the four arcs need?
The same width from both crossings, and more than half the circle's diameter.
How are the compass width and the radius of the circle connected?
The width you set is the radius of the circle you draw with it.
How many arcs do you draw, and from where?
Four. Two from each crossing, one on each side of the line.
What do you join to finish?
The two points where the arcs cross. That line is the perpendicular through P.
Which shape do the equal arcs and the shortened segment form?
A rhombus, with the shortened segment as one of its diagonals.
How do you construct a perpendicular through a point that is NOT on the line?
Centre the circle on the point, big enough to cross the line (or its extension) twice, then construct as before.
How can you check your finished construction?
Measure the angle between the line and your perpendicular with a protractor. It should be close to 90 degrees.
What is the perpendicular distance from a point to a line?
The distance along the perpendicular from the point to the line. It is the shortest distance from the point to the line.
A point's perpendicular lands only on the extension of a segment. What is the shortest distance to the segment?
The distance to the nearer endpoint.

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.