TransformationsEdexcel GCSE Maths: Revision notes

A transformation is a process that moves or changes a shape. There are four main types you must understand:

Reflections, rotations and translations all preserve congruence – this means the shape stays the same size and shape. Only enlargement changes the size.

When describing any transformation, you must be precise and complete – examiners will not give full marks for vague descriptions.

A reflection creates a mirror image of a shape across a line called the mirror line (or line of reflection).

To reflect a shape:

1. Identify the mirror line (e.g. the x-axis, y-axis, or a line like y = x)
2. For each point on the original shape, measure its perpendicular distance to the mirror line
3. Mark the reflected point the same distance on the opposite side of the line
4. Join the reflected points to create the image

To describe a reflection, you must state the mirror line explicitly. For example:

• "Reflection in the x-axis"
• "Reflection in the line y = 1"
• "Reflection in the line x = -2"

Common mirror lines in exam questions:

• The x-axis (y = 0)
• The y-axis (x = 0)
• The line y = x
• Vertical lines like x = 2
• Horizontal lines like y = -1

A rotation turns a shape around a fixed point called the centre of rotation.

To rotate a shape, you need three pieces of information:

1. Centre of rotation (a fixed point, often the origin)
2. Angle of rotation (usually 90°, 180°, or 270°)
3. Direction (clockwise or anticlockwise)

To rotate a shape:

1. Choose a point on the shape
2. Draw a line from the centre of rotation to that point
3. Rotate this line by the given angle in the specified direction
4. The distance from the centre stays the same
5. Mark the new position and repeat for all points

Complete descriptions must include all three elements:

• "Rotation of 90° anticlockwise about the origin"
• "Rotation of 180° about the point (2, 1)"
• "Rotation of 270° clockwise about (0, 0)"

Key fact: A rotation of 180° is the same whether clockwise or anticlockwise.

A translation slides a shape from one position to another. Every point on the shape moves the same distance in the same direction.

To translate a shape, use a vector which shows:

• How far to move right/left (horizontal component)
• How far to move up/down (vertical component)

Vectors are written as $\begin{pmatrix} a \\ b \end{pmatrix}$ where:

• a = number of units right (positive) or left (negative)
• b = number of units up (positive) or down (negative)

Examples of vector notation:

• $\begin{pmatrix} 3 \\ 2 \end{pmatrix}$ means 3 units right, 2 units up
• $\begin{pmatrix} -2 \\ 5 \end{pmatrix}$ means 2 units left, 5 units up
• $\begin{pmatrix} 4 \\ -3 \end{pmatrix}$ means 4 units right, 3 units down

To describe a translation, state the vector:

• "Translation by vector $\begin{pmatrix} 3 \\ 2 \end{pmatrix}$"

If you can identify a point on the original shape and its image, you can work out the vector by calculating the horizontal and vertical changes.

An enlargement changes the size of a shape. You must specify a centre of enlargement and a scale factor.

Scale factor rules:

• Scale factor > 1: Shape gets larger
• 0 < Scale factor < 1: Shape gets smaller (fractional scale factor)
• Negative scale factor: Shape is enlarged and rotated 180° about the centre

To enlarge a shape:

1. Identify the centre of enlargement
2. For each point, draw a line from the centre through the point
3. Multiply the distance from centre to point by the scale factor
4. Mark the new position

Example: If a shape is enlarged by scale factor 2 from the origin, every distance from the origin doubles.

Critical: Enlargement does NOT preserve congruence – the image is a different size.

How area and perimeter change:

• If scale factor is k, then all lengths are multiplied by k
• Perimeter is multiplied by k
• Area is multiplied by k²

Examples:

• Scale factor 2: perimeter × 2, area × 4
• Scale factor 3: perimeter × 3, area × 9
• Scale factor 0.5: perimeter × 0.5, area × 0.25
• Scale factor -2: perimeter × 2, area × 4 (direction of scale factor ignored for area/perimeter)

In exam questions, you may be given two shapes and asked to describe the single transformation that maps one to the other.

Strategy: Decide which type of transformation it is

1. Check if shapes are congruent (same size and shape)

   • If yes → reflection, rotation, or translation
   • If no → enlargement

2. If congruent, determine which of the three:

   • Reflection? One shape is a mirror image; find the mirror line (check if it's equidistant from corresponding points)
   • Rotation? Find the centre (point that doesn't move), measure the angle, state direction
   • Translation? All points move the same distance in the same direction; work out the vector

3. If not congruent → it's an enlargement

   • Find the scale factor: (length in image) ÷ (length in object)
   • Find the centre of enlargement by extending lines through corresponding points

Always describe completely:

• Name the transformation
• State all required information (line for reflection, centre/angle/direction for rotation, vector for translation, centre/scale factor for enlargement)
• Do not just write 'translated' – write 'translated by vector $\begin{pmatrix} 2 \\ -3 \end{pmatrix}$'