Transformations
Transformations
A transformation maps every point of a shape to a new position. The four transformations at GCSE are translation, reflection, rotation and enlargement. For each, you must be able to both perform the transformation (given a description, draw the image) and describe it fully (given original and image, state the transformation and all its parameters).
Four Transformations — What to State
| Transformation | Image size | To describe fully, state: |
|---|---|---|
| Translation | Same | The vector \( \begin{pmatrix} x \ y \end{pmatrix} \) |
| Reflection | Same | The mirror line (equation or description) |
| Rotation | Same | Centre, angle, direction (clockwise / anticlockwise) |
| Enlargement | Different | Centre of enlargement and scale factor |
Key Points
- Translation, reflection and rotation produce congruent images — size is preserved.
- Enlargement (scale factor \( k \neq 1 \)) produces a similar image — shape is preserved but size changes.
- A fractional scale factor (\( 0 < k < 1 \)) produces a smaller image.
- To find the centre of rotation: the perpendicular bisector of any two corresponding points passes through the centre.
- To find the centre of enlargement: draw lines through each pair of corresponding points — they all meet at the centre.
Worked Examples
Triangle \( A \) has vertices \( (1,1),(3,1),(3,4) \). Triangle \( B \) has vertices \( (1,-1),(3,-1),(3,-4) \). Describe the single transformation.
Reflection in the \( x \)-axis (the line \( y=0 \)).
Triangle \( A \) maps to triangle \( C \) at \( (-1,1),(-3,1),(-3,4) \). Describe the transformation.
Reflection in the \( y \)-axis (the line \( x=0 \)).
Triangle with vertices \( (2,2),(6,2),(6,8) \) is enlarged by scale factor \( \frac{1}{2} \) from the origin. Find the image vertices.
\[ (2,2)\to(1,1) \qquad (6,2)\to(3,1) \qquad (6,8)\to(3,4) \]Each coordinate is multiplied by the scale factor when the centre is the origin.
Point \( P(5,3) \) is rotated 90° anticlockwise about the origin. Find the image.
The rule for 90° anticlockwise rotation about the origin: \( (x,y)\to(-y,x) \).
\[ (5,3) \to (-3,5) \]Key Takeaways
- Translation: vector only. Reflection: mirror line. Rotation: centre + angle + direction. Enlargement: centre + scale factor.
- Translation, reflection, rotation → congruent (isometric). Enlargement → similar.
- Coordinate rotation rules about the origin: 90° ACW: \( (x,y)\to(-y,x) \). 180°: \( (x,y)\to(-x,-y) \). 90° CW: \( (x,y)\to(y,-x) \).
- Fractional scale factor: enlargement but image is smaller than original.