Transformations

Topic Overview

In mathematics, 'transforming' an object mean that you change its appearance. During your GCSE maths exam, you will be required to recognise and perform four different types of transformation on various shapes.

The four types of transformations which you will encounter during this topic are:

  • Rotation
  • Reflection
  • Translation
  • Enlargement/Re-sizing

With the exception of enlargement/ re-sizing, transformations such as rotation, reflection and translation will not alter the size of the shape. After these transformations, the shape will still have exactly the same size, area, angle sizes and line lengths.

When transforming a shape, the original shape is called 'object' and the transformed shape is called the 'image'.

If the shape in question has been rotated, reflected or translated, it will remain the same size. As a result, the transformed image will be 'congruent' to the original object.

However, if the shape in question is enlarged or re-sized, it will change in size. As a result, the transformed image will be 'similar' to the original object.

For more information on congruence and similarity, see the 'Congruent Triangles' revision guide.

Key Concepts

In the new linear GCSE Maths paper, you will be required to solve various transformation questions. The specific questions you will be expected to answer will vary depending upon which examination board with which you are registered, but as a rule you will be required to:

  • Perform translations of various 2D shapes using; rotations, translations, reflections, enlargements/resizing
  • Perform enlargements of 2D shapes using both positive and negative scale factors
  • Recognise the properties of various transformations

Listed below are a series of summaries and worked examples to help you solidify your knowledge about the different types of transformations.

Worked Examples

1. Translations
If an object is 'translated' this means that it is moved up, down, left or right of its original position on a grid. When translating an object, its shape size and direction remain the same.

When describing a translation, you must use displacement vectors. These are written in the form:

\[(\frac{a}{b})\]

where 'a' describes how many places the object has been moved left or right, and 'b' describes how many places the object has been moved upwards or downwards.

When writing displacement vectors, the horizontal displacement is always written at the top of the vector and the vertical displacement is always written at the bottom.

A downwards translation or a translation to the left is displayed using a minus sign.

For example:

\[(\frac{4}{{ - 3}})\]

means that an object has been translated (moved) 4 places to the right and 3 places downwards, whereas

\[(\frac{-4}{{3}})\]

would mean that an object has been translated 4 places to the left and 3 places upwards.

Example
(a) - Write the displacement vector which describes triangle P's translation to triangle Q:

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Solution
(a) - By counting the squares in the grid or by counting the numbers on the axes, you can see that, in order to be translated to triangle Q, triangle P has moved 1 place downwards and 4 places to the right.

Therefore, the displacement vector for the translation is,

\[(\frac{4}{{ - 1}})\]

1. Reflections
If an image is 'reflected', this means that it faces the opposite direction of the original object.

Every point of the original object and new image is the same distance from the central line between them. The central line is often referred to as the 'mirror line', as the object and its image 'mirror' one another.

In a reflection, the object and its image are always the same perpendicular distance from this mirror line. For example:

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Points A and E, B and D, as well as C and F are the same perpendicular distance from the mirror line, which in this example is the 'y' axis.

In order to reflect an object, you follow these steps: 1. Measure from the point to the mirror line, 2. Measure the same distance again on the other side and place a dot, 3. Connect these new dots together to make a reflection of the original object.

Example
(a) - Sketch the mirror line in which the L-shape ABCDEF has been reflected.

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Solution
(a) - In order for the L shape ABCDEF to have been reflected, each original point and its reflected image must be the same distance from the mirror line. Therefore, you must count the number of squares between L shape ABCDEF and its reflection, and then sketch a line directly down the middle. You answer should look like this:

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3. Rotations
If an object is ‘rotated’, this means that it is turned a specific number of degrees around a centre point. The diagram below demonstrates some of the most common degrees of image rotation:

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The general rule for rotation of an object 90° anti-clockwise or 270° clockwise is:

\[(x,y) \to ( - y,x)\]

The general rule for rotation of an object 180° is:

\[(x,y) \to ( - x,-y)\]

The general rule for rotation of an object 270° anti-clockwise or 90° clockwise is:

\[(x,y) \to ( y,-x)\]

If an object is rotated 360°, it returns to its original position, i.e.

\[(x,y) \to (x,y)\]

This is referred to as a 'full rotation'.

When rotating an object, the distance between the centre and any point on the object must stay the same. Moreover, every point on the object makes a circular movement around the centre point.

Example
(a) - Describe the rotation which has occurred to the red triangle in the following diagram:

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Solution

(a) - When rotating an object, the distance between the centre and any point on the object must stay the same. Therefore, in order to find the centre point of rotation, you must find a point which is the same distance between the red and blue triangle; which in this example is the point (0,0).

In order to work out the degree of rotation, it is necessary to look at the coordinates of the original object and its new image:

The red triangle has coordinates: (1,1), (4,1) and (2,4)

The blue triangle has coordinates (-1, -1), (-4, -1) and (-2, -4)

The general rule for rotation of an object 180° is (x, y) → (-x, -y).

Therefore, the red triangle has been rotated 180° about the centre point (0,0).

4 - Enlargements/Resizing
If you 'enlarge' or 'resize' an object, this means that you change its size but keep its original shape the same. In order to carry out enlargements, you need to know the scale factor of the object and the centre of enlargement.

The 'scale factor' tells us how much the object has been enlarged or resized.

The 'centre of enlargement' tells us the specific point from which the enlargement is being measured.

Example
(a) - Enlarge the following image by a scale factor of 2.5 from the central point of enlargement (0,1)

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Solution
(a) - In order to enlarge the following image, you must follow this process: Firstly, locate the central point of enlargement (0,1) and draw lines from this point which pass through the 4 corners of the rectangle.

Secondly, count the number of places between the centre of enlargement (0,1) and each of the points on the rectangle.

Your scale factor is 2.5, so you must multiply the distance between the centre point of enlargement and each of your co-ordinates by 2.5 respectively.

Count this number of places from the centre point of enlargement and mark these 4 new points with dots.

Finally, connect all of these dots and you will have correctly enlarged the rectangle by scale factor 2.5 from the centre point of enlargement (0,1). Your new image should look like this:

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During your GCSE maths exam, you may also be required to resize various objects. This is a similar process to enlargement, except you may be required to reduce the original object's size rather than increase it:

Example
(b) - Resize the rectangle WXYZ by a scale factor of -2, centred about the origin:

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Solution
(b) - In order to resize this object, you must follow a similar process to enlargement:

Firstly, locate the central point of resizing (0,0) and draw lines from this point which pass through the 4 corners of the rectangle.

Secondly, count the number of places between the centre of resizing (0,0) and each of the points on the rectangle.

Your scale factor is -2, so you must multiply the distance between the centre point of enlargement and each of your co-ordinates by -2 respectively.

Count this number of places from the centre point of enlargement and mark these 4 new points with dots.

Finally, connect all of these dots and you will have correctly resized the rectangle by scale factor -2 from the centre point of resizing (0,0). Your new image should look like this:

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(Note: During the process of resizing, you will notice that the object is on the other side to the centre of enlargement, and the object has been turned upside down).

Exam Tips

  1. Use a sharp pencil when performing transformations in order to improve the accuracy of your answers.
  2. Label your answers correctly: note down the 'x' and 'y' axes, the points of both your original object, the new image, and the centre point of transformation. This will display your understanding of transformations to the examiner who is marking your paper and earn you maximum method marks.
  3. Remember that, with the exception of enlargement/ resizing, transformations such as rotation, reflection and translation will not alter the size of the shape. After these transformations, the shape will still have exactly the same size, area, angle sizes and line lengths.
  4. When writing displacement vectors, remember that the horizontal displacement is always written at the top of the vector and the vertical displacement is always written at the bottom, and that a downwards translation or a translation to the left is displayed using a minus '-' sign.
  5. If an image is 'reflected', this means that it faces the opposite direction of the original object.
  6. Memorise the following rules for rotations: if an object is rotated 90° anti-clockwise or 270° clockwise: (x, y) → (-y, x), if an object is rotated 180°: (x, y) → (-x, -y), if an object is rotated 270° anti-clockwise or 90° clockwise: (x, y) → (y, -x), and if an object is rotated 360°, it returns to its original position, i.e. (x, y) → (x, y). This is referred to as a 'full rotation'.
  7. Bear in mind that if you 'enlarge' or 'resize' an object, this means that you change its size but keep its original shape the same. In order to carry out an enlargement or resizing, you need to know the scale factor of the object and the centre of enlargement.

Topic Summary

When answering exam questions which involve transformations, make sure that you have the following equipment:

  • A sharpened pencil
  • A ruler
  • A rubber
  • A pencil sharpener
  • Spare pencils

If you make sure that you bring this equipment to your GCSE maths exam, you will improve the accuracy of your answers, as well as being able to correct any mistakes easily and clearly.

Moreover, it is important that you practice as many transformations past paper questions as possible. If you can gain experience at recognising and performing a variety of different transformations, you will increase your confidence and subsequently improve your chances of scoring maximum marks for this topic during your actual GCSE maths exam.

Related Topics

  • Polygons
  • Congruent Triangles
  • Trigonometry and Pythagoras
  • Construction Problems and Loci
  • Perimeter, Area, Volume
  • Circle Theorem Problems
  • Proof