One to one maths interventions built for KS4 success

Weekly online one to one GCSE maths revision lessons now available

Learn more

Transformations

Here we will learn about transformations, including similar shapes and congruent shapes and triangles, reflections, translations,  rotations and enlargements.

There are also transformations worksheets based on Edexcel, AQA and OCR exam questions, along with further guidance on where to go next if you’re still stuck.

What are transformations?

Transformations change the size and/or the position of a shape.

To do this we need a 2D shape (such as a polygon) and to follow the instructions given.  These instructions are sometimes known as a mapping.

There are four geometric types of transformations:

Reflection

Reflections involve a mirror line, also known as a line of reflection. 

E.g.

Transformations Image 1

The shapes are congruent.

Step-by-step guide: Reflection

Rotations

Rotations involve a centre of rotation, an angle of rotation and a direction of rotation (clockwise or anticlockwise).

E.g.

Transformations Image 2

The shapes are congruent.

Step-by-step guide: Rotations

Translation

Translations involve a move in a horizontal direction and a move in a vertical direction.

E.g.

Transformations Image 3

The shapes are congruent.

Step-by-step guide: Translation

Enlargement

Enlargements make a shape bigger or smaller. They must have a scale factor and they may involve a centre of enlargement

E.g.

Transformations Image 4

The shapes are similar.

Questions can involve a single transformation or at Higher, GCSE questions can combine transformations.

When we combine transformations we need to do the first transformation on the original shape, then carry out a second transformation on the new shape. 

Step by step guide: Enlargement

What are transformations?

What are transformations?

What are scale factors?

Scale factors tell us how much bigger or smaller a shape will become when it is enlarged.

To enlarge a shape we multiply each side length of the shape by the scale factor.

They can be integers or fractions.

  • If the scale factor is greater than 1, the shape gets larger.
    E.g.
    The scale factor of enlargement from shape A to shape B is 2
Transformations Image 5

  • If the scale factor is between 0 and 1, the shape gets smaller.
    E.g.
    The scale factor of enlargement from shape A to shape B is \frac{1}{2} or 0.5
Transformations Image 6

At GCSE Higher, there may be a negative scale factor of enlargement.

Step-by-step guide: Scale factor

What are invariant points?

Invariant points are points which have stayed in the same place after a transformation.

E.g.

Here is a reflection.  The invariant point is labelled.

Transformations Image 7

Similar and congruent shapes

For geometrical transformations we need to understand the terms similar and congruent.

  • Similar shapes

Similar shapes are two or more shapes which are the same shape, but different sizes. One shape is an enlargement of the other.

E.g.

These two triangles are similar.

Transformations Image 8

Step-by-step guide: Similar shapes

  • Congruent shapes

Congruent shapes are shapes that are exactly the same.

E.g.

These two triangles are congruent.

Transformations Image 9

Step-by-step guide: Congruent shapes

  • Congruent triangles

Congruent triangles also have special conditions of congruence. These involve looking at their sides and angles. 

The four conditions are known by abbreviations:

SSS (three sides the same),

RHS (right-angled triangle, hypotenuse and a side the same),

ASA or AAS (two angles and one side the same),

SAS (side-angle-side, two sides and the included angle the same).

Transformations Image 10

Step-by-step guide: Congruent triangles

How to use transformations

  • Use a pencil, additional construction lines and tracing paper to help.
  • Remember:
    Reflections have a mirror line.
    Rotations have an angle of rotation, a direction and a centre of rotation.
    Enlargements have a scale factor (and sometimes a centre of enlargement).
    Translations have a distance across and a distance up or down.
  • Similar shapes are linked to enlargements.
  • Congruent shapes are linked to reflection, rotations and translations.

Transformations worksheet

Transformations worksheet

Transformations worksheet

Get your free transformations worksheet of 20+ questions and answers. Includes reasoning and applied questions.

DOWNLOAD FREE
x
Transformations worksheet

Transformations worksheet

Transformations worksheet

Get your free transformations worksheet of 20+ questions and answers. Includes reasoning and applied questions.

DOWNLOAD FREE

Transformations examples

Example 1: similar shapes

These shapes are similar. Find the value of x.

Transformations Example 1

  • Match up the corresponding sides and consider the ratio.

    4:x=7:14

  • Find a scale factor.

    14\div 7 =2

  • Use the scale factor to find the missing side.

    x=4\times 2=8

    The missing side is 8 cm .

Example 2: congruent shapes

Which shape is congruent to shape A?

Transformations Example 2

  • Look at the shapes and see if any are the same type of shape.

  • Then look carefully and check the corresponding angles and sides. Congruent shapes can be rotated or reflected or both.

  • Shape Y is congruent to shape A.

Example 3: congruent triangles

Are these triangles congruent?

Transformations Example 3

  • Check the corresponding sides and angles.

    \begin{aligned} &\text{side} \ AC = \text{side} \ DE \\ &\text{angle} \ BAC = \text{angle} \ EDF\\ &\text{angle} \ BCA = \text{angle} \ FED \end{aligned}

  • Triangle ABC is congruent to triangle DEF.
    This is because they have two equal corresponding angles and the included side in both triangles is also equal. This is known as angle-side-angle (ASA).

Example 4: reflection

Reflect the shape in the mirror line.

Transformations Example 4

  • Consider each point (the vertices of the original shape) and reflect it in the mirror line. The new points will be the same distance from the mirror line as the original points.
Transformations Example 4-1

  • Then join up the new points.
Transformations Example 4-2

Example 5: rotations

Rotate the shape 90^{\circ} clockwise about the centre of rotation.

  • Use tracing paper.  Trace the original shape and check where the centre of rotation is.
Transformations Example 5

  • Put the tip of your pencil on the centre of rotation.  Rotate the correct angle and in the right direction.
Transformations Example 5-1

  • Then draw your new shape onto the paper.
Transformations Example 5-3

Example 6: translation

Translate the shape using the vector

\begin{pmatrix} 3\\ -1 \end{pmatrix}​

Transformations Example 6

  • Choose a point (vertex) of the original shape and move it using the column vector.

  • The column vector gives instructions on how to move the original point.

    \begin{pmatrix} 3\\ -1 \end{pmatrix} \text{is} \begin{pmatrix} 3 \ \text{right}\\ 1 \ \text{down}\\ \end{pmatrix} ​

Transformations Example 6-1

  • Repeat for the other points (vertices). Or you can just draw in the rest of the polygon. You may use tracing paper to check.
Transformations Example 6-2

  • Then join up the new points.
Transformations Example 6-3

Example 7: enlargements – scale factor

Enlarge the shape by scale factor 3

Transformations Example 7

  • Choose a side to enlarge first.  The left-side in the original shape is 1 , so the left-side in the new shape will be 9 .  This can be drawn anywhere on the grid.

    3\times3=9
Transformations Example 7-1

  • Enlarge the other sides using the same scale factor.
Transformations Example 7


The lengths of the sides of the second shape are three times the lengths of the original shape.

Example 8: enlargements – centre of enlargement

Enlarge the shape by scale factor 2 , using the centre of enlargement.

Transformations Example 8

  • Draw ray lines through the centre of enlargement and each of the vertices of the original shape.
Transformations Example 8-1

  • Measure the distance from the centre of enlargement to a point (vertex) and then use the scale factor of 2 and mark the position of the new point.
Transformations Example 8-2

  • Repeat for the other points (vertices).
Transformations Example 8-3

  • Then join up the new points.
Transformations Example 8-4

Example 9: combined transformations (higher)

Reflect triangle A in the line x=1 to give triangle B.

Rotate triangle B is rotated 180^{\circ} about O to give triangle C.

Transformations Example 8 (Higher)

  • Carry out the first transformation on the original shape (triangle A).
Transformations Example 8 (Higher)-1

  • Then carry out the second transformation on the new shape (triangle B). O is the origin, (0,0) .
Transformations Example 8 (Higher)-2

You may be asked to describe the single transformation that maps triangle A onto triangle C.

For this example the single transformation would be:

Rotate triangle A 180^{\circ} about (1,0) to give triangle C.

Common misconceptions

  • Use the grid to help put the vertices of the shape in the correct position

Geometric transformations of shapes are often on grid paper.  Use the lines and squares on the grid to help you be accurate about the position of the new shape.

  • You can add extra details onto your diagram to help you

Try not to do transformation work in your head.  Make use of your pencil and ruler to add marks to help you.  You may also use tracing paper.

  • Describe transformations fully

When you are asked to describe a transformation make sure you state which kind of transformation it is and all other details.

  • Make sure you have the object and the image the right way around

The original shape is the object and the translated shape is the image.  Make sure you know which shape is the original shape and start there when describing transformations such as translations.

Practice transformations questions

1. These two shapes are similar. Find the missing value of x .

 

Transformations Practice Question 1

x=12
GCSE Quiz False

x=18
GCSE Quiz True

x=14
GCSE Quiz False

x=15
GCSE Quiz False

Sides BC and CE are a pair of corresponding sides. Sides AC and CD are also a pair of corresponding sides.

 

\begin{aligned} BC:CE&=AC:CD\\\\ 4:12 \; &= \; 6:x \end{aligned}

 

The scale factor is

 

12\div4=3

 

x=6\times3=18

2. Reflect the shape in the mirror line.

 

Transformations Practice Question 2

 

Transformations Practice Question Answer 2

GCSE Quiz False

Transformations Practice Question Answer 3

GCSE Quiz False

Transformations Practice Question Answer 1

GCSE Quiz True

Transformations Practice Question Answer 4

GCSE Quiz False

Reflect each point in the mirror line. The new points will be the same distance from the mirror line as the original points.

 

Transformations Practice Question 2 Explanation

3. Rotate the shape 90^{\circ} anti-clockwise about the point O.

 

Transformations Practice Question 3

 

Transformations Practice Question 3 Answer 1

GCSE Quiz True

Transformations Practice Question 3 Answer 2

GCSE Quiz False

Transformations Practice Question 3 Answer 3

GCSE Quiz False

Transformations Practice Question 3 Answer 4

GCSE Quiz False

Use tracing paper to help with the rotation.

 

Transformations Practice Question 3 Explanation Image

4. Translate the shape using the vector

 

\begin{pmatrix} -1\\ 4 \end{pmatrix}

 

Transformations Practice Question 4

 

Transformations Practice Question 4 Answer 2

GCSE Quiz False

Transformations Practice Question 4 Answer 5

GCSE Quiz False

Transformations Practice Question 4 Answer 6

GCSE Quiz False

Transformations Practice Question 4 Answer 1

GCSE Quiz True

 The column vector gives the instructions on how to move each of the points.

 

\begin{pmatrix} -1\\ 4 \end{pmatrix} \; \text{is} \;\; \begin{aligned} &1 \ \text{left}\\ &4 \ \text{up}\\ \end{aligned} ​

5. Enlarge the shape using the scale factor \frac{1}{2}

 

Transformations Practice Question 5

 

Transformations Practice Question 5 Answer 2

GCSE Quiz False

Transformations Practice Question 5 Answer 1

GCSE Quiz True

Transformations Practice Question 5 Answer 3

GCSE Quiz False

Transformations Practice Question 5 Answer 4

GCSE Quiz False

The scale factor is \frac{1}{2} . The side lengths of the original shape need to be halved (or divided by 2 ). The height of the original shape is 4 , so the height of the new shape will be 2 .

 

4\times \frac{1}{2}=2

 

The base of the original shape is 2 , so the base of the new shape will be 1.

 

2\times \frac{1}{2}=1

6. Enlarge the shape by scale factor 3 , using the centre of enlargement O.

 

Transformations Practice Question 6

 

Transformations Practice Question 6 Answer 2

GCSE Quiz False

Transformations Practice Question 6 Answer 3

GCSE Quiz False

Transformations Practice Question 6 Answer 1

GCSE Quiz True

Transformations Practice Question 6 Answer 4

GCSE Quiz False

The distances between the points of the original shape and the centre of enlargement need to be multiplied by three.

 

\begin{aligned} &OP=2\\ &OP’=2\times 3=6 \end{aligned}

 

Transformations Practice Question 6 Explanation Image

Transformations GCSE questions

1. Give a reason why the two triangles are congruent.

 

Transformations GCSE Question 1 Transformations GCSE Question 1-1

 

(1 mark)

Show answer

Side-side-side (SSS)

(1)

2. Triangles ABC and DEF are similar.

 

Transformations GCSE Question 1-2

 

Work out the value of x

 

(2 marks)

Show answer
12\div4=3

For the scale factor

(1)

x=3\times3=9

For the correct value of x

(1)

3.  (a)       Translate shape A using the vector

 

\begin{pmatrix} 4\\ -1 \end{pmatrix} ​

 

Transformations GCSE Question 3

 

(b) Describe the single transformation that maps shape A onto shape P.

 

Transformations GCSE Question 3b

 

(4 marks)

Show answer

(a)

 

Transformations GCSE Question 3a

(2)

 

(b)

 

Transformations GCSE Question 3b-1

 

Reflection

(1)

In the line x=4

(1)

4 (higher). Triangle A is rotated 90^{\circ} clockwise about the point (0, 0) to give triangle B.

 

Triangle B is translated by the vector \begin{pmatrix} 3\\ 2 \end{pmatrix} to give triangle C.

 

Transformations GCSE Question 4

 

Describe fully the single transformation that maps triangle A onto triangle C.

 

(3 marks)

Show answer

Transformations GCSE Question 4-1

 

Rotation

(1)

180^{\circ}

(1)

About (1.5,2)

(1)

Learning checklist

You have now learned how to:

  • Recognise congruent shapes and triangles
  • Find missing lengths of similar shapes using a scale factor
  • Reflect shapes in a mirror line
  • Rotate shapes about a point
  • Translate shapes using a column vector
  • Enlarge shapes using a scale factor
  • Enlarge shapes with a centre of enlargement

Still stuck?

Prepare your KS4 students for maths GCSEs success with Third Space Learning. Weekly online one to one GCSE maths revision lessons delivered by expert maths tutors.

GCSE Benefits

Find out more about our GCSE maths revision programme.

x

FREE GCSE Maths Practice Papers - 2022 Topics

Practice paper packs based on the advanced information for the Summer 2022 exam series from Edexcel, AQA and OCR. 



Designed to help your GCSE students revise some of the topics that will come up in the Summer exams.

Download free