Transformations
A transformation moves or resizes a shape. There are four kinds at GCSE, and each needs particular details before a description is complete. Most lost marks here come from a missing detail rather than a wrong drawing.
Part 1 of 3: Learn it
In short
- Translation: give the column vector. Reflection: give the mirror line.
- Rotation: give the angle, the direction and the centre.
- Enlargement: give the scale factor and the centre.
Where this is in your specification
Spec points: DfE G7 and G24 (negative scale factors are Higher tier)
| Board | Topic: Transformations, congruence and similarity |
|---|---|
| DfE content | G5, G7, G8, G19, G24 |
| AQA 8300 | Geometry and measures |
| Edexcel 1MA1 | Geometry and measures |
| OCR J560 | 8 Basic geometry; 9 Congruence and similarity |
| Eduqas C300 | HG5, HG7, HG8, HG19 |
| Cambridge IGCSE 0580 | E4.4, E4.5, E7.1 |
| National 5 C847 75 | Geometric skills: similarity |
The four transformations
| Transformation | What it does | Details for a full description |
|---|---|---|
| Translation | slides the shape | a column vector, such as (3 over −2) |
| Reflection | flips the shape in a mirror line | the equation of the line, such as y = x |
| Rotation | turns the shape | angle, direction (clockwise or anticlockwise), centre |
| Enlargement | changes the size | scale factor and centre |
Translations, reflections and rotations give congruent shapes (same size and shape). Enlargements give similar shapes (same shape, different size).
Reflections and mirror lines
- x = a is a vertical line, y = b is a horizontal line.
- Reflecting in the x-axis changes the sign of y: (4, 1) goes to (4, −1).
- Reflecting in y = x swaps the coordinates: (4, 1) goes to (1, 4).
- Reflecting in y = −x swaps them and changes both signs: (4, 1) goes to (−1, −4).
Rotations
Use tracing paper: trace the shape, hold the centre still with a pencil point, turn the paper by the angle and draw the new position. About the origin, 90° clockwise sends (x, y) to (y, −x), and 180° sends (x, y) to (−x, −y).
Enlargements
Every distance from the centre is multiplied by the scale factor. A scale factor between 0 and 1 makes the shape smaller. With a negative scale factor (Higher), the image is on the opposite side of the centre and upside down.
Lengths are multiplied by the scale factor k, but areas are multiplied by k².
Reflect (6, −2) in the y-axis.
Show the answer
(−6, −2).
Part 2 of 3: See it worked
Worked examples
Example 1
Translate the point (3, −1) by the vector (−5 over 4).
- x: 3 + (−5) = −2
- y: −1 + 4 = 3
Answer: (−2, 3).
Example 2
Enlarge the point (2, 4) by scale factor 3, centre (1, 2).
- From the centre to the point: 1 right, 2 up
- Times 3: 3 right, 6 up
- From (1, 2): (1 + 3, 2 + 6)
Answer: (4, 8).
Common mistakes
- Describing a rotation without its centre or direction.
- Mixing up x = 2 (vertical) and y = 2 (horizontal) as mirror lines.
- Measuring an enlargement from the origin when the centre is somewhere else.
- Writing a vector as a coordinate, such as (3, −2), when a column vector is needed.
Enlarge (8, 4) by scale factor ½ with centre the origin.
Show the answer
(4, 2).
Part 3 of 3: Test yourself
Check yourself
Answer each one in your head or on paper first, then open it to check.
Reflect (6, −2) in the y-axis.
(−6, −2).
Enlarge (8, 4) by scale factor ½ with centre the origin.
(4, 2).
A shape is enlarged by scale factor 3. What happens to its area?
It is multiplied by 3² = 9.
Jobs that use this
Each link opens the job profile on the National Careers Service (England). In the rest of the UK: My World of Work (Scotland), Careers Wales, nidirect careers (Northern Ireland).
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