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Geometry and Measurement · Shape and Space · Lesson 4

Constructing Perpendicular Lines with Compasses and Ruler

A builder checking that a wall stands straight, a farmer marking rows at right angles to a path, and a surveyor setting out a plot all need perpendicular lines. In this lesson you will use only a pair of compasses and a ruler to copy a line segment, draw a perpendicular at a point on a line, and drop a perpendicular from a point outside a line.

NaCCA B7.3.1.2.1 ⏱ about 50 min

By the end you will be able to

  • ✔Copy a given line segment using compasses and a ruler.
  • ✔Construct a perpendicular to a line at a point on the line.
  • ✔Construct a perpendicular from a point outside a line to the line.
  • ✔Explain why each construction works and describe the steps in order.

Key words

Perpendicular
Two lines are perpendicular if they meet at a right angle (90°).
Compasses
An instrument with a point and a pencil, used to draw circles and arcs of a fixed radius.
Arc
A part of a circle, drawn with compasses.
Radius
The distance between the compass point and the pencil; the distance from the centre to the arc.
Line segment
The part of a line between two end points, with a definite length.
Curriculum reference

Content standard: B7.3.1.2 Demonstrate how to construct a perpendicular to a line from a given point, bisect a line, bisect angles, and construct angles of the following sizes: 30°, 45°, 60°, 75° and 90°.

Indicator: Construct a line segment perpendicular to another line segment.

Starter

The farmer's right-angled rows

Mr Yaw, a vegetable farmer near Ejura, has a straight footpath along the edge of his field. He wants his rows of cabbages to start at the path and run exactly at right angles to it. He has no set square, only a piece of string, two sticks and a ruler. With the string as 'compasses' he can mark points on the path and find the line at 90° to it. The same idea is used on paper with a pair of compasses, and it works every time.

1. Tools and the idea behind constructions

A construction is a drawing made with only a ruler (a straight edge for drawing lines, not for measuring angles) and a pair of compasses. We do not use the protractor. Because the compasses draw points that are an exact distance from a centre, a construction can be more accurate than measuring.

Before you start, check your tools. Use a sharp pencil so that the lines are thin. Make sure the compasses do not slip: tighten the screw, so the radius stays the same while you draw. Always leave your construction arcs visible. They are your working and earn marks.

  1. 1Draw lightly at first and leave all the arcs on the paper.
  2. 2Never change the compass setting between two arcs that must have the same radius.
  3. 3Label every point with a capital letter.
  4. 4Finish by drawing the final line firmly and mark the right angle with a small square.

2. Copying a line segment

The simplest construction is to copy a line segment, because it uses the main skill of compasses: remembering a length. Suppose you must copy the segment PQ. Draw a line and mark a point R on it. Place the compass point on P and open it until the pencil is on Q. Without changing the opening, put the compass point on R and draw an arc that cuts the line at a point S. Then RS = PQ.

▶ The arc from R keeps the same radius as the opening PQ, so RS has exactly the same length as PQ.

This idea of 'same radius means same distance' is used in every construction that follows.

3. A perpendicular at a point on a line

Task: Given a line AB and a point P on it, construct a line through P perpendicular to AB.

  1. 1Place the compass point on P. With any convenient radius, draw two arcs that cut the line on each side of P, at X and Y. Now PX = PY.
  2. 2Open the compasses to a bigger radius (more than PX). Place the point on X and draw an arc above the line.
  3. 3Keeping the same radius, place the point on Y and draw another arc that crosses the first one. Call the crossing Z.
  4. 4Use the ruler to join P to Z. Then PZ is perpendicular to AB, so ∠ZPX = ∠ZPY = 90°.

▶ Follow the arcs: two arcs on the line, two crossing arcs above, then the line through P.

Why does it work? P is exactly in the middle of XY, and Z is the same distance from X as from Y. Any point equally far from X and Y lies on the line that cuts XY at right angles through its middle. P is the middle, so PZ is that line.

4. A perpendicular from a point outside the line

Task: Given a line AB and a point P not on it, drop a perpendicular from P to the line. Imagine P is a well and AB is a straight road. The perpendicular is the shortest path from the well to the road.

  1. 1With the compass point on P, draw an arc that cuts the line AB in two places, X and Y.
  2. 2Place the compass point on X and draw an arc on the side of the line opposite to P, with a radius more than half of XY.
  3. 3With the same radius, place the point on Y and draw another arc to cut the first one at Z.
  4. 4Join P to Z with a ruler. The line PZ cuts AB at a point F, and PF is perpendicular to AB.

▶ Arcs from P cut the line twice; two matching arcs from those points meet at a new point. The line through P and that point is perpendicular.

The distance from P to F is the shortest distance from P to the line AB. Every other point on the road is further from the well. This is why a pipe laid straight from the well to the road is shorter than one that goes in a slant.

Three constructions side by side
ConstructionStarting pointKey idea
Copy a segmenta segment and a point Rsame compass opening, same length
Perpendicular at a pointP on the lineequal arcs either side of P, then crossing arcs
Perpendicular from a pointP off the linearc from P cuts the line, then equal crossing arcs

Worked examples

Try each one yourself first, then open the solution and compare every step.

Example 1

Copy a line segment

Segment PQ is 6.5 cm long. Describe how to construct a segment RS equal to PQ.
Show the solution
  1. Step 1Draw a line with the ruler and mark a point R on it.
  2. Step 2Place the compass point on P and open the compasses until the pencil is exactly on Q.
  3. Step 3Keep the opening fixed. Put the point on R and draw an arc to cut the line at S.
  4. Step 4Measure RS: it should be 6.5 cm, the same as PQ.

Answer: RS = 6.5 cm = PQ

Example 2

Perpendicular at a point

Describe the construction of a line perpendicular to AB at the point P on AB.

▶ Two equal arcs on the line, two equal crossing arcs, and then the perpendicular.

Show the solution
  1. Step 1With centre P draw arcs that cut AB at X (left of P) and Y (right of P).
  2. Step 2With centre X and a larger radius, draw an arc above AB.
  3. Step 3With centre Y and the same radius, draw an arc to cross it at Z.
  4. Step 4Join P to Z. Angle ZPY = 90°; mark it with a square.

Answer: PZ is perpendicular to AB at P

Example 3

Perpendicular from an outside point

P is a point 4 cm above a line AB. Describe how to construct the perpendicular from P to AB.

▶ The perpendicular from P meets the line at F, forming 90°.

Show the solution
  1. Step 1With centre P draw an arc that cuts AB at X and Y.
  2. Step 2With centre X draw an arc below the line (radius more than half XY).
  3. Step 3With centre Y and the same radius, draw an arc to cut the first at Z.
  4. Step 4Join P to Z. It crosses AB at F. PF ⟂ AB.
  5. Step 5Measure PF: this is the shortest distance from P to AB.

Answer: PF is perpendicular to AB and is the shortest distance from P to AB

Example 4

Using the perpendicular distance on a map

On a map, Kofi finds the perpendicular distance from a school to a highway to be 3.6 cm. The scale is 1 cm to 50 m. What is the real shortest distance?
Show the solution
  1. Step 1Real distance = map distance × scale.
  2. Step 23.6 × 50 = 180.
  3. Step 3So the real distance is 180 m.

Answer: 180 m

Example 5

Counting the angles made

A perpendicular is constructed at P on a straight line. Each of the two right angles is then bisected. How many angles of 45° appear, and what do they add up to?
Show the solution
  1. Step 1The perpendicular splits the 180° straight angle into two angles of 90°.
  2. Step 2Bisecting each 90° angle gives two angles of 45°, so in total 4 angles of 45°.
  3. Step 3Total: 4 × 45° = 180° ✓.

Answer: 4 angles of 45°, adding to 180°

Summary

  • Constructions use only compasses and a ruler; keep all construction arcs visible.
  • To copy a segment: set compasses to its length, then draw an arc from the new starting point.
  • To construct a perpendicular at a point P on a line: equal arcs either side of P, then two equal crossing arcs, then join.
  • To drop a perpendicular from P outside a line: arc from P cutting the line twice, then two equal arcs crossing on the other side, then join.
  • The perpendicular from a point to a line is the shortest distance from the point to the line.

Practice: 15 questions

5 easy, 5 medium, 5 hard. Each one is marked straight away, with a full worked solution.

You can practise without an account, but sign up free or sign in to save your progress.

1 easy
Perpendicular lines meet at an angle of:
2 easy
Which instrument is used to draw an arc of a fixed radius?
3 easy
A perpendicular is drawn to a line at a point P. What angle does it make with the line, in degrees?
°
4 easy
What do we call the shortest line from a point to a line? It meets the line at 90°.
5 easy
To construct a perpendicular at a point P on a line, what do you do first?
6 medium
When you draw the two crossing arcs from X and Y, the radius must be:
7 medium
Segment PQ is 7.4 cm long. A copy RS is constructed. How long is RS, in centimetres?
cm
8 medium word problem
Kofi stands at a well and measures the distance from it to a straight road. The shortest distance he finds is 5.2 m. What is the length of the perpendicular from the well to the road, in metres?
m
9 medium
Put the steps for dropping a perpendicular from P in the correct order: (1) draw arcs from X and Y to meet at Z; (2) join P to Z; (3) arc from P cutting the line at X and Y.
10 medium
A perpendicular XP is constructed at P on a straight line AB, so ∠APX = 90°. The angle XPB is then bisected. Find the size of each of the two new angles, in degrees.
°
11 hard word problem
Dzifa constructs a perpendicular at a point P on a line AB. The arcs centred on P cut the line at X and Y, with PX = PY = 3 cm.

  1. (a) What is the length of XY, in centimetres?

  2. (b) The crossing arcs from X and Y must have a radius greater than what length, in centimetres?

  3. (c) What is the smallest whole number of centimetres she can use as the radius of the crossing arcs?

12 hard
Which statement about constructing a perpendicular from a point P to a line is NOT correct?
13 hard word problem
On a map, the perpendicular distance from a school to a highway measures 3.6 cm. The scale is 1 cm to 50 m. Find the real shortest distance from the school to the highway, in metres.
m
14 hard word problem
A perpendicular is constructed at P on a straight line, and both right angles are then bisected. How many angles of 45° appear?
15 hard
Select ALL the true statements about constructing a perpendicular at a point on a line.

Choose all that are correct.