Geometric Construction
Introduction

Making accurate drawings is not a skill confined to artists. Draftsmen, architects, engineers, and designers need to be able to construct geometrical figures.

In this unit we will look at how to carry out some common geometric constructions using only a pair of compasses and a ruler.
Equidistance
In the map shown in Fig.1 below, the town of Richley is 5 km from both Bridgeford and from Adleborough. We say that Richley is equidistant from Bridgeford and from Adleborough.
Figure 1.   Three towns marked on a map.
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When we are told only that a place is say 8 km away from us we do not know in what direction it lies, so we do not know its exact position. However, from this information we can find all the possible positions of the place. If we plot a
point
  • A point has no properties except position. It is an object with zero dimensions.
  • Points in the x-y plane can be specified using x and y coordinates.
point
the same distance away in every possible direction, we find we have drawn a circle.

Click somewhere in the animation space below to set a distance from the starting point and see what happens.
Figure 2.   The possible positions of a point a given distance away.
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If you were told that Ceyton was
equidistant
Items that are at an equal distance from an identified point, line or plane are said to be equidistant from it.
equidistant
from Ayton and Beeville, and you were given the following map, can you see where the possible positions of Ceyton would be? The circles represent all the points a given distance from Ayton and all the points the same distance away from Beeville. Try dragging the green handle to alter the size of the red circles (they should always be the same size as one another).
Figure 3.   Map of the Ayton area.
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Select the points below that could represent the position of Ceyton. (Hint: use the circles to check whether each point is equidistant from Ayton and Beeville.)
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The intersections of the two congruent (same-sized) circles represent the possible positions of Ceyton – points that are equidistant from Ayton and Beeville. Adjusting the
radius
A straight line segment joining the centre of a circle with a point on its circumference (or the length of this line).
radius
of the circles shows that there are lots of points where Ceyton could be.

We can simplify the map and use a geometric model. Click on the button to plot all the points that are equidistant from A and B (shown in red).

Figure 4.   Geometric model of the Ayton area.
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We can see that all the points that are equidistant from A and B make up a
straight line
A straight line is a set of points related by an equation of the form y = ax + c. It has length and position, but no breadth and is therefore one-dimensional.
straight line
. We say that this line is the locus of all the points equidistant from A and B. Fig.5 below shows how we would construct such a line.

Figure 5.   How to draw a line equidistant from two points.
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Constructing perpendicular lines
We are still unsure of the exact position of Ceyton, although we know that it lies along a line of equidistance. There is, however, some extra information about it in a guide book: 'The picturesque town of Ceyton lies five miles due south of the aptly named Halfway Motel. This pleasant motel is located exactly halfway along the straight road joining Ayton and Beeville.' So the Halfway Motel, represented by the point H, must lie as shown below.

Figure 6.   Updated map of Ayton area.
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We can see that H is the
midpoint
The midpoint of a line is the point halfway along it.
midpoint
of AB and it lies at the intersection of line 1 and AB. So in fact line 1 bisects, or cuts in half, the line AB.

Not only this, but line 1 is
perpendicular
Two lines or planes are perpendicular if they are at right angles to one another.
perpendicular
to the
line segment
A line segment is the set of points on the straight line between any two points, including the two endpoints themselves.
line segment
AB. We therefore call line 1 the perpendicular bisector of the line
segment
The region of a circle bounded by an arc and the chord joining its two end points.
segment
AB.

So when we need to draw the
perpendicular bisector
A perpendicular bisector is a line that cuts in half a given line segment and forms a 90° angle with it.
perpendicular bisector
of a line segment, we can use the same method as we used in Fig.5 before.

Figure 7.   How to draw a perpendicular bisector.
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Practice this construction on paper. Use a straight edge or ruler to draw a line segment of about 5 cm. Copy the process shown in Fig.7 above to
bisect
To bisect something is to cut it in half. In mathematics, lines and angles are often bisected.
bisect
your line segment.

Sometimes you may be asked to draw a perpendicular line through a particular point, rather than halfway along the line. Our technique can be adjusted to tackle such a construction:

Figure 8.   How to construct a perpendicular line through a given point.
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Now, draw a line on your paper and label it 'line 2'. Place a point P anywhere on the line. You should be able to draw a line perpendicular to line 2 that contains P using the technique shown above.

Constructing an equilateral triangle
We cannot directly measure whether an
angle
An angle is a measure of turning. Angles are measured in degrees. The symbol for an angle is .
angle
is 60 using just a ruler and compasses. However, compasses can ensure that distances are constant, and as long as the three side lengths of a
triangle
A triangle is a three-sided polygon.
triangle
are equal, then the three angles will be equal. Because the angles in a triangle always sum to 180°, each of the angles must therefore be 60°.

Figure 9.   A triangle with equal length sides has angles of 60°.
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Once we have created a line segment to form the base of the triangle, this defines the length of the other two sides.

Figure 10.   How to construct an equilateral triangle.
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Now use a ruler and compasses to recreate on paper the pattern in Fig.11 below using the constructions you have learnt so far.

Figure 11.   A pattern made from equilateral triangles.
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Bisecting an angle
Look at the triangle AOM in Fig.12 below. We can make another triangle by reflecting triangle AOM in the line OM as if it were a mirror. Click on the button to see this new triangle. You should notice that point B is the mirror image of point A, so the new triangle is BOM.

Figure 12.   Constructing a mirror image of a triangle.
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From Fig.12 we can see that is equal to . This means that:





So is half of . We have shown that the line OM bisects .

Line OM also cuts something else in half. As line segments AM and BM are equal in length, we can see that OM bisects line segment AB. Also, we know from the original diagram that , so the line segment AB is perpendicular to line segment OM. Putting all this information together, we can see that line OM is the perpendicular bisector of line segment AB.

We can use this knowledge to construct a line that bisects a given angle. Fig.13 below shows how to do this, first by finding two points, like A and B in Fig.12, and then by constructing the perpendicular bisector of the line segment between them.

Figure 13.   How to bisect an angle.
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Constructing parallel lines
Parallel
Two lines, curves or planes are said to be parallel if the perpendicular distance between them is always the same.
Parallel
lines are lines that exist in the same
plane
A plane has position, length and width but no height. It is an object with two-dimensions.
plane
but do not
intersect
To intersect is to have a common point or points. For example, two lines intersect at a point and two planes intersect at a straight line. The point at which two or more lines intersect is called a vertex.
intersect
. The perpendicular distance between straight
parallel
Two lines, curves or planes are said to be parallel if the perpendicular distance between them is always the same.
parallel
lines is always constant. Move the
vertical
A vertical line runs at a right angle to a horizontal line. The vertical axis of a graph runs from top to bottom of a page.
vertical
measuring line in the simulation below to see this.


Click on the figure below to interact with the model.

 Figure 14.  Two parallel lines are a constant perpendicular distance from one another.



So, if we find two points the same perpendicular distance away from a given line and on the same side of it, then the line through these two points will be parallel to the original line.

Most often we will need to construct a parallel line through a given point that is some distance away from the original line. This is how it is done:

Figure 15.   How to draw a parallel line through a given point.
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Copying an angle
Copying an angle (and the lines that make it) can be done by recreating distances using compasses. We start by taking one of the angle's lines as a base, copying its length using the compasses, and then creating another line of exactly the same length. Watch how the whole process is carried out:
Figure 16.   How to copy an angle.
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Now try a similar construction yourself.

Use a straight edge, such as a ruler, to draw a triangle on your paper, leaving enough space to make a copy of it. Label the vertices A, B and C.

Now use the above method to copy . Finally, by drawing in one more straight line, you should be able to turn your construction into a copy of the full triangle.

What is the additional line that you must draw to complete the copied triangle?
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Summary

If a point A is equidistant from two other points B and C, the distance AB is equal to the distance AC.

In geometric constructions, compasses are used to reproduce equal lengths.

Compasses can be used to draw two points D and E that are equidistant from the two points B and C. A ruler can then be used to draw a straight line connecting D and E. This is the perpendicular bisector of the line segment BC.

Given an original line a parallel line can be drawn by constucting a line through two points that are the same perpendicular distance away from the initial line.

An angle of 60 can be constructed by using compasses to create three vertices a set distance from each other. A ruler can then be used to connect the corners and create the required angle.

An angle formed by two lines can be bisected by using compasses of a fixed size to mark off two points on the lines, each an equal distance from the angle. The perpendicular bisector of the line segment between these two points bisects the original angle.

An angle within a triangle can be copied by using compasses to copy each of the sides of the triangle.
Exercises
1. When arranged in the correct order, the statements below form instructions on how to construct an
equilateral triangle
An equilateral triangle has three sides of the same length and hence three angles of 60°.
equilateral triangle
. Match each statement to its position in the instructions (1 to 5).
  • Label one of the points where the circles intersect C.
    Draw a line segment AB using a ruler.
    Set your compasses to the length AB.
    Using your ruler draw the line segments AC and AB
    Lightly draw a circle, centre A, and a circle, centre B using the compasses.
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2. Which of the following can be constructed just using a ruler and a pair of compasses?
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