Transformations

This lesson covers the four geometric transformations for AQA GCSE Mathematics: reflection, rotation, translation and enlargement. You need to be able to perform each transformation and, just as importantly, describe a transformation fully. Questions on transformations are very common and often carry 2-3 marks each.

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The Four Transformations

graph TD
    A[Transformations] --> B[Reflection]
    A --> C[Rotation]
    A --> D[Translation]
    A --> E[Enlargement]
    B --> B1[Mirror line]
    C --> C1[Centre, angle, direction]
    D --> D1[Column vector]
    E --> E1[Centre, scale factor]

Transformation	What Changes?	What Stays the Same?
Reflection	Orientation (flipped)	Size and shape (congruent)
Rotation	Position, orientation	Size and shape (congruent)
Translation	Position	Size, shape, orientation (congruent)
Enlargement	Size (and position)	Shape and angles (similar)

Exam Tip: Reflections, rotations, and translations all produce congruent images (same shape and size). Only enlargement changes the size of the shape, producing a similar image.

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Reflection

To perform a reflection, you need a mirror line (line of reflection).

Rules:

• Each point on the image is the same perpendicular distance from the mirror line as the original point, but on the opposite side.
• The mirror line is the perpendicular bisector of the line segment joining each point to its image.

Common Mirror Lines

Mirror Line	Equation	Description
x-axis	y = 0	Horizontal line through the origin
y-axis	x = 0	Vertical line through the origin
y = x	y = x	Diagonal line through the origin at 45 degrees
y = -x	y = -x	Diagonal line through the origin at -45 degrees

Worked Example 1

Reflect the point (3, 2) in the line y = x.

Solution: When reflecting in y = x, swap the coordinates: (3, 2) becomes (2, 3).

How to Describe a Reflection

You must state:

1. The transformation is a reflection
2. The equation of the mirror line

Example: "A reflection in the line x = -1."

Exam Tip: When describing a reflection, you MUST give the equation of the mirror line (e.g., "y = 2" or "x = -1"), not just "the vertical line" or "the horizontal line." Without the equation, you will lose marks.

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Rotation

To perform a rotation, you need three pieces of information:

1. Centre of rotation — the fixed point about which the shape turns
2. Angle of rotation — how far it turns (usually 90 degrees or 180 degrees)
3. Direction — clockwise or anticlockwise (not needed for 180 degrees)

Worked Example 2

Rotate triangle ABC with vertices A(1, 1), B(3, 1), C(1, 3) by 90 degrees clockwise about the origin.

Solution: For a 90-degree clockwise rotation about the origin, apply the rule: (x, y) becomes (y, -x).

• A(1, 1) becomes (1, -1)
• B(3, 1) becomes (1, -3)
• C(1, 3) becomes (3, -1)

How to Describe a Rotation

You must state:

1. The transformation is a rotation
2. The centre of rotation
3. The angle of rotation
4. The direction (clockwise or anticlockwise) — unless 180 degrees

Example: "A rotation of 90 degrees anticlockwise about the point (0, 0)."

Finding the Centre of Rotation

To find the centre of rotation between a shape and its image:

1. Connect a point on the original to its corresponding point on the image.
2. Draw the perpendicular bisector of this line segment.
3. Repeat for another pair of corresponding points.
4. The centre of rotation is where the perpendicular bisectors intersect.

Exam Tip: If you are asked to describe a rotation and you leave out the centre, the angle, or the direction, you will not get full marks. All three (or two for 180 degrees) are required.

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Translation

A translation moves every point of a shape by the same distance in the same direction. It is described using a column vector.

A column vector is written as two numbers stacked vertically:

• Top number: horizontal movement (positive = right, negative = left)
• Bottom number: vertical movement (positive = up, negative = down)

Worked Example 3

Translate triangle PQR by the column vector (3 on top, -2 on bottom) where 3 means 3 right and -2 means 2 down.

If P = (1, 4), then the image P' = (1 + 3, 4 + (-2)) = (4, 2).

How to Describe a Translation

You must state:

1. The transformation is a translation
2. The column vector

Example: "A translation by the vector (4 on top, -1 on bottom)," meaning 4 right and 1 down.

Exam Tip: You MUST use a column vector to describe a translation. Writing "4 right and 1 down" in words will not get full marks at GCSE. Practise writing column vectors neatly.

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Enlargement

An enlargement changes the size of a shape. You need:

1. Centre of enlargement — the fixed point from which the shape is enlarged
2. Scale factor — how much bigger (or smaller) the image is

Types of Scale Factor

Scale Factor	Effect
Greater than 1 (e.g. 2, 3)	Shape gets larger
Between 0 and 1 (e.g. 1/2, 1/3)	Shape gets smaller (a reduction)
Negative (e.g. -2) [H]	Shape is enlarged and inverted (on the other side of the centre)

How to Perform an Enlargement

1. Draw lines (ray lines) from the centre of enlargement through each vertex of the shape.
2. Multiply the distance from the centre to each vertex by the scale factor.
3. Plot the new vertices at these distances along the ray lines.

Worked Example 4

Enlarge triangle with vertices A(2, 1), B(4, 1), C(2, 3) by scale factor 2, centre of enlargement (0, 0).

Solution: Multiply each coordinate by 2:

• A(2, 1) becomes (4, 2)
• B(4, 1) becomes (8, 2)
• C(2, 3) becomes (4, 6)

Worked Example 5 (Fractional Scale Factor)

Enlarge a shape by scale factor 1/2, centre (0, 0). Point P = (6, 4).

Solution: P' = (6 x 1/2, 4 x 1/2) = (3, 2)

Worked Example 6 (Negative Scale Factor) [H]

Enlarge point Q(3, 1) by scale factor -2, centre of enlargement (1, 1).

Solution: Vector from centre to Q = (3-1, 1-1) = (2, 0) Multiply by -2: (-4, 0) Image Q' = (1 + (-4), 1 + 0) = (-3, 1)

How to Describe an Enlargement

You must state:

1. The transformation is an enlargement
2. The scale factor
3. The centre of enlargement

Example: "An enlargement by scale factor 3, centre (2, -1)."

Finding the Centre of Enlargement

Draw lines through corresponding vertices of the shape and its image. The point where all lines meet is the centre of enlargement.

Finding the Scale Factor

Scale factor = (distance from centre to image point) / (distance from centre to original point)

Or equivalently: scale factor = (length of image side) / (length of original side)