Constructions and Tilings

Chapter Overview

Geometric construction means drawing shapes using precise rules rather than measurement by eye.

The main tools are:

• ruler or straightedge;
• compass;
• pencil.

1. Perpendicular Bisector

Construction

For segment $AB$:

1. open the compass to more than half of $AB$;
2. draw arcs from $A$ above and below the segment;
3. with the same opening, draw arcs from $B$;
4. join the two intersection points of the arcs.

The new line is the perpendicular bisector of $AB$.

2. Constructing a Right Angle

A perpendicular line gives an angle of:

So a perpendicular bisector construction can also be used to create a right angle at the midpoint of a segment.

3. Angle Bisector

Construction

For $\angle ABC$:

1. draw an arc centred at $B$ cutting both arms;
2. from those two points, draw equal arcs that intersect;
3. join $B$ to the intersection point.

The new ray bisects the angle.

4. Copying an Angle

To copy an angle:

1. draw a new starting ray;
2. draw an arc across the original angle;
3. copy the same arc at the new point;
4. transfer the distance between the original arc intersections;
5. draw the second ray through the marked point.

5. Constructing 60°

An equilateral triangle has three equal angles.

Since:

each angle is:

Therefore constructing an equilateral triangle provides a simple construction of a $60^\circ$ angle.

6. Constructing Parallel Lines

If a transversal cuts two lines and corresponding angles are equal, the lines are parallel.

We can use this fact:

1. draw a line and a point outside it;
2. draw a transversal through the point;
3. copy the corresponding angle at the outside point;
4. extend the new ray.

The new line is parallel to the original line.

7. Tilings

Common shapes that can tile the plane by themselves include:

• squares;
• equilateral triangles;
• regular hexagons.

8. Why Compass Constructions Work

A compass creates sets of points at a fixed distance from a centre. This simple fact explains many constructions.

When equal-radius arcs are drawn from the two ends of a segment, their intersection points are equally distant from both endpoints. The line through those intersections is therefore the perpendicular bisector.

9. Perpendicular Bisector Property

Every point on the perpendicular bisector of segment $AB$ is equally distant from $A$ and $B$.

If point $P$ lies on the perpendicular bisector, then:

This property can also be used in reverse: if a point is equally distant from $A$ and $B$, it lies on the perpendicular bisector of $AB$.

10. Angle Bisector Property

An angle bisector divides an angle into two equal angles.

If ray $OP$ bisects $\angle AOB$, then:

The construction works by creating equal distances with compass arcs.

11. Constructing 60° and Related Angles

An equilateral triangle has three $60^\circ$ angles. This provides a standard construction for $60^\circ$ using equal-radius arcs.

Once $60^\circ$ and angle bisection are available, other angles can be constructed. For example, bisecting $60^\circ$ gives $30^\circ$.

12. Parallel Lines by Construction

One way to construct a line parallel to a given line is to copy a corresponding angle. If a transversal makes equal corresponding angles with two lines, the lines are parallel.

13. Understanding Tilings

A tiling covers a surface with shapes without gaps and without overlaps. At a meeting point, the angles around the point must fit exactly into:

For regular polygons, this helps explain which shapes can tile by themselves.

For example, a square has interior angle $90^\circ$:

so four squares fit exactly around a point.

An equilateral triangle has angle $60^\circ$:

so six equilateral triangles can fit around a point.

14. Symmetry and Repetition in Tilings

Tilings often involve translations, rotations and reflections of the same shape. Recognising the repeating unit helps us analyse and design patterns efficiently.

15. Complete Construction Notes

Tools and Their Roles

A ruler is used to draw straight lines and segments. A compass transfers equal distances and draws arcs or circles. A protractor may measure angles, but classical ruler-and-compass constructions depend on geometric properties rather than measuring every angle directly.

Constructing a Perpendicular Bisector

For segment $AB$:

1. open the compass to more than half of $AB$;
2. draw arcs above and below the segment centred at $A$;
3. without changing the radius, repeat from $B$;
4. join the two arc intersections.

The resulting line cuts $AB$ at its midpoint and at $90^\circ$.

Constructing an Angle Bisector

For $\angle AOB$:

1. draw an arc centred at $O$ meeting both arms;
2. from those two intersection points, draw equal arcs that meet inside the angle;
3. join $O$ to the new arc intersection.

The new ray divides the original angle into two equal parts.

Copying an Angle

Copying an angle uses the same arc radius in the original and new location, then transfers the chord distance between the two arc intersections. This reproduces the original opening exactly.

Constructing Parallels

A line can be made parallel to another by creating equal corresponding or alternate angles with a transversal. The construction is therefore connected directly to the angle relationships from the parallel-lines chapter.

Tilings with Regular Polygons

For a regular polygon to tile by itself around a point, its interior angle must fit into $360^\circ$ a whole number of times.

Regular hexagon interior angle is $120^\circ$:

so three regular hexagons can meet at a point.

Squares and equilateral triangles also tile regularly. Regular pentagons do not tile the plane by themselves using the same simple arrangement.

Mixed Tilings

Different polygons can sometimes meet around a point if their angles total exactly $360^\circ$. For example, combinations of triangles and squares can create repeating tilings.

Common Mistakes

Quick Revision

Practice Questions

Apply Your Learning

Using a ruler and compass, construct:

1. a perpendicular bisector of a segment;
2. a $60^\circ$ angle;
3. its angle bisector to obtain $30^\circ$.

Keep the construction arcs visible.

Additional Practice

Chapter Summary