Transformations: Reflection, Rotation, Translation & Enlargement

GCSE Maths · Geometry

Transformations

A transformation changes the position, size or orientation of a shape. There are four types at GCSE.

Translation

A translation slides a shape without rotating or resizing it. Described by a column vector.

The vector (a, b) means move a units right (negative = left) and b units up (negative = down).

Worked Example: Translate triangle A by the vector (3, −2).

Every point moves 3 right and 2 down
If a vertex is at (1, 5), it moves to (4, 3)

Reflection

A reflection creates a mirror image across a line of reflection.

Common mirror lines:

x = a (vertical line)
y = b (horizontal line)
y = x (diagonal, 45°)
y = −x (diagonal, −45°)

Key property: Each point and its image are the same perpendicular distance from the mirror line.

Worked Example: Reflect the point (3, 1) in the line x = 5.

The point is 2 units left of x = 5
The image is 2 units right of x = 5: (7, 1)

Describing a reflection: You must state "reflection" AND the equation of the mirror line.

Rotation

A rotation turns a shape through a given angle about a fixed centre of rotation.

To describe a rotation, state:

The word "rotation"
The angle (e.g. 90°, 180°)
The direction (clockwise or anticlockwise) — not needed for 180°
The centre of rotation (a coordinate)

Worked Example: Rotate the point (3, 1) by 90° clockwise about the origin (0, 0).

Rule for 90° clockwise about origin: (x, y) → (y, −x)
(3, 1) → (1, −3)

Useful rules for rotation about the origin:

90° clockwise: (x, y) → (y, −x)
90° anticlockwise: (x, y) → (−y, x)
180°: (x, y) → (−x, −y)

Enlargement

An enlargement changes the size of a shape from a centre of enlargement by a scale factor.

Scale factor > 1: shape gets bigger
0 < scale factor < 1: shape gets smaller (but it is still called an enlargement)
Negative scale factor (Higher): shape is enlarged AND inverted (appears on the opposite side of the centre)

Method: From the centre of enlargement, multiply the distance to each vertex by the scale factor.

Worked Example: Enlarge triangle with vertices (2, 1), (4, 1), (2, 3) by scale factor 2, centre (0, 0).

(2, 1) → (4, 2)
(4, 1) → (8, 2)
(2, 3) → (4, 6)

Worked Example (Higher): Enlarge by scale factor −2, centre (1, 1).

For point (3, 2):

Vector from centre to point: (3−1, 2−1) = (2, 1)
Multiply by −2: (−4, −2)
New point: (1−4, 1−2) = (−3, −1)

Describing Transformations

When asked to describe a transformation, you must give all required information:

Transformation	Must State
Translation	Vector
Reflection	Mirror line equation
Rotation	Angle, direction, centre
Enlargement	Scale factor, centre

Properties Preserved

	Translation	Reflection	Rotation	Enlargement
Shape	✓	✓	✓	✓ (similar)
Size	✓	✓	✓	✗
Orientation	✓	✗	✗	✓

Exam Tips

Missing ANY part of a description loses marks — e.g. "rotation 90°" without the centre or direction is incomplete
Use tracing paper in the exam to check rotations and reflections
For enlargements, count squares from the centre to find new positions
A "single transformation" means exactly ONE — do not describe it as two steps

Transformations: Reflection, Rotation, Translation & Enlargement