Geometry · GCSE Maths
Transformations
GCSE Maths transformations: translation by vector, reflection in a mirror line, rotation (centre, angle, direction), and enlargement (centre, scale factor positive, negative or fractional).
Name the transformation, then give its full description: vector for translation; mirror line; centre, angle and direction for rotation; centre and scale factor for enlargement.
The important bits
What you need to know
- 1
Translation: every point moves the same vector (x over y). Shape, size and orientation stay the same. On a grid, count right then up from a vertex.
- 2
Reflection: flip across a mirror line. Each point is the same perpendicular distance on the other side of the line. State the equation of the mirror line, e.g. x = 2 or y = x.
- 3
Rotation: turn about a centre through an angle (90°, 180°, 270°) clockwise or anticlockwise. Trace paper helps; name centre, angle and direction.
- 4
Enlargement: rays from the centre through each vertex. Scale factor k multiplies distances from the centre. k > 1 grows; 0 < k < 1 shrinks.
- 5
Negative scale factor: enlargement on the opposite side of the centre, with size multiplied by |k|. Orientation is reversed (upside down).
- 6
Fractional scale factor ½: half the size, same side of centre as positive enlargement. Centre of enlargement is where the rays meet.
- 7
Combined transformations: order matters. Reflect then translate is generally not the same as translate then reflect.
- 8
Describe fully — “rotation” alone is incomplete. Examiners want centre, angle and direction, or mirror line, or vector, or centre and scale factor.
Quotations worth analysing
Short evidence. Real method.
“Translation by vector (3 over −2)”
Top number is x (right positive); bottom is y (up positive). (−2) means 2 down. Every vertex moves by the same vector.
“Enlargement, centre O, scale factor −2”
Twice the size, on the far side of O from the object. The negative flips orientation; the 2 doubles length.
“Reflection in the line y = x”
Swap coordinates: (a, b) maps to (b, a). Perpendicular distance to the line is preserved.
Go deeper
Translation and reflection on the grid
Translate triangle ABC by (4 over 1): move every vertex 4 right, 1 up. A(1, 2) → A′(5, 3). Check all three vertices the same way. Reflect in x = 3: each point’s x-coordinate is mirrored — distance from x = 3 is preserved. Point (5, 4) is 2 right of the line, so image is (1, 4). Reflect in y = x: swap x and y. Reflect in the x-axis: (x, y) → (x, −y). Always draw the mirror line lightly if it is not given. Describing a reflection: “reflection in the line y = −x” not “flipped”. Construction marks: right angles to the mirror line through each vertex, equal distances.
Go deeper
Rotation with centre not at the origin
Rotate 90° clockwise about (2, 1): place tracing paper pin at (2, 1), trace the shape, turn clockwise 90°, transfer points. Without tracing paper: draw horizontal and vertical through the centre, rotate each vertex relative to the centre. 90° clockwise about the origin: (x, y) → (y, −x). 180° about any centre reverses position through that centre. State “90° clockwise about (2, 1)” — missing the centre loses marks. Direction matters: 90° clockwise ≠ 90° anticlockwise except for symmetric shapes about the centre.
Go deeper
Enlargement: positive, fractional and negative k
Enlargement centre (0, 0), scale factor 3: multiply each coordinate by 3. (2, 1) → (6, 3). Scale factor ½: multiply coordinates by ½ — closer to the origin. Scale factor −2 about (0, 0): multiply by −2, so (1, 2) → (−2, −4) — opposite side of origin, double size, inverted. To find the centre: draw rays through corresponding vertices on object and image; they meet at the centre. Fractional negative enlargements appear on Higher papers — same ray method, |k| for size, negative for side flip. Describe as “enlargement, centre (…), scale factor …”.
See the idea in action
Translate point P(−1, 4) by the vector (3 over −2). State the image coordinates. Step 1: Translation adds the vector to each coordinate: x′ = −1 + 3, y′ = 4 + (−2). Step 2: x′ = 2, y′ = 2. Step 3: Image P′ is (2, 2). Check: moved 3 right (from −1 to 2) and 2 down (from 4 to 2), matching the vector.
Exam technique
Turn knowledge into marks
When describing, give the complete specification in one sentence: e.g. “rotation 90° anticlockwise about (3, −1)”. Use tracing paper in the exam if allowed.
Common mistakes
Do not give these marks away
- 01
Describing a transformation without the mirror line, vector, centre, angle, direction or scale factor.
- 02
Mixing up 90° clockwise and anticlockwise coordinate rules, or rotating about the origin when the centre is elsewhere.
- 03
Treating a negative scale factor as “smaller” — |k| controls size; the sign controls which side of the centre the image lies.
A shape is mapped by enlargement, centre (0, 0), scale factor 2. Point (3, 5) goes to
A(6, 10)
B(5, 7)
C(1.5, 2.5)
D(−6, −10)
Show the answer
(6, 10). Multiply both coordinates by 2: (6, 10). (1.5, 2.5) would be scale factor ½. (−6, −10) would be scale factor −2.
Quick questions
If this is the bit you searched
How do I describe a translation fully?
State “translation by vector (a over b)” with the column vector in the form the board expects. Do not say “moved right” without the vector.
What does a negative scale factor mean?
The image lies on the opposite side of the centre from the object, with lengths multiplied by |k|. Orientation is reversed.
How do I find the centre of enlargement?
Draw rays through matching vertices on the object and image. The point where the rays meet is the centre.
Does order matter for combined transformations?
Yes. Reflect then translate usually gives a different result from translate then reflect. Follow the order stated in the question.