“Shape & Space” or geometry is a big subject – so much so that we can talk about different types of geometry depending what we are focusing on.

Here we are going to look at a relatively modern part of geometry called transformational geometry.

Transformational geometry is interested in taking a shape and changing it in some way. The specific changes that we look at are: translations, rotations, reflections and enlargements.

Let’s start with reflections:

Reflections

To reflect a shape in a mirror line, we need to take every vertex of the shape to the same distance on the other side of the mirror line (let’s see what this means). If we then join up all of our new vertices we will have the image of our original shape after a reflection. Be careful – the mirror line won’t always be horizontal or vertical! (Although it will always be straight during the iGCSE course)!

Practice together

Exercise

Now let’s complete exercise 15A on pages 234 to 236 of the textbook:

The answers are below:

Now let’s turn our attention to rotations:

Rotations

A rotation turns a shape around a certain point. We must remember that there is always a specific point that is the centre of rotation. From than point every part of the shape rotates by a specified angle (e.g. 90º or 180º) in a specified direction.

Practice together

Let’s try some:

More Examples

Exercise

Let’s try questions 2 to 4 from exercise 15B on pages 236 to 237 of the textbook:

The answers are below:

Now let’s look at translations:

Translations

A translation always has a horizontal component which moves to the right (or the left) and a vertical component which move up (or down). We use vector notation to specify this translation, putting the horizontal translation on the top and the vertical translation on the bottom.

Let’s try doing this first with individual points in space. Then we can try doing it with complete shapes. Let’s also try identifying what vector a point or shape has been translated by.

Practice together

Let’s try identifying the translation vectors for a point that has moved and marking the image point following a translation ourselves, and then let’s try the examples below where we are dealing with a whole shape:

Exercise

Let’s complete exercise 15C on pages 239 and 240 of the textbook:

The answers are below:

Combining different transformations

Questions will often want you to identify which transformation has taken place without telling you what kind of transformation you are looking for. They may also ask you to do several transformations one after another.

Exercise

In the short exercise, exercise 15D on page 241, we will practice applying more than one geometrical transformation:

The answers are below:

Now let’s look at a slightly more complicated type of transformation called an enlargement.

Enlargements

If we need to identify the centre and scale factor of an enlargement and we know the object (the original shape) and the image, we can use a special method where we draw a continued line joining each of the corresponding vertices of the object and the image. Where these lines meet will be the centre, and the relative distances from the centre will give us the scale factor of the enlargement. This is easier to understand by seeing it done.

To “do” an enlargement from a given centre, we draw a continued line from the centre to each of the vertices and then mark a point “n” times further along that line, where n is our scale factor. Again, let’s try this first with a drawing.

Exercise

Now let’s try our last exercise in the topic of geometric transformations, which is questions 2 to 5 of exercise 15E on pages 243 to 244 of the textbook:

The answers are below: