Reflection: Flipping Shapes Across a Line
Reflection produces a mirror image of a shape by flipping it across a straight line called the line of reflection.
Each point on the original shape and its reflected image are the same perpendicular distance from the line of reflection, but on opposite sides.
The shape's size and orientation relative to the line change, but the shape itself remains congruent to the original (same size and shape).
- Line of reflection is a straight line.
- Distance from each point to the line equals distance from image point to the line.
- Orientation reverses (e.g., clockwise becomes anticlockwise).
Rotation: Turning Shapes Around a Centre
Rotation turns a shape about a fixed point called the centre of rotation.
The angle of rotation is measured in degrees and must be a multiple of 90° (90°, 180°, 270°, or 360°) for this syllabus.
The shape maintains its size and shape but changes orientation depending on the angle and direction (usually anticlockwise).
- Centre of rotation is a fixed point.
- Angle of rotation is a multiple of 90°.
- Orientation changes but size and shape remain the same.
- Example: Rotating a triangle 90° anticlockwise around a point moves each vertex accordingly.
Enlargement: Scaling Shapes from a Centre
Enlargement changes the size of a shape by scaling it from a fixed centre point.
The scale factor determines how much the shape grows or shrinks:
Positive scale factors greater than 1 enlarge the shape (make it bigger). Example: scale factor 2 doubles all distances from the centre, so the shape doubles in size but keeps the same shape and orientation (no rotation). Positive scale factors between 0 and 1 reduce the shape (make it smaller). Example: scale factor 0.5 halves all distances from the centre. Negative scale factors also change the size but include a rotation of 180° about the centre, effectively producing an inverted image on the opposite side of the centre point.
- Centre of enlargement is fixed.
- Scale factor can be positive, fractional, or negative.
- Negative scale factor includes a 180° rotation.
- Shape remains similar (same shape, different size).
- Example: Enlarging a square by scale factor -1.5 from centre O produces a square 1.5 times larger, rotated 180° about O.
Translation: Sliding Shapes Along a Vector
Translation moves every point of a shape the same distance in the same direction, defined by a vector.
A vector has two components: horizontal (x) and vertical (y) shifts.
The shape does not change size, shape, or orientation during translation.
- Vector defines direction and distance.
- All points move equally along the vector.
- Shape remains congruent and oriented the same way.
- Example: Translating a triangle by vector (3, -2) moves every vertex 3 units right and 2 units down.
Combining Transformations
Transformations can be applied one after another to produce more complex effects.
The order of transformations matters; changing the order can produce different results.
When combining, carefully track each step and apply transformations in sequence.
- Apply transformations sequentially.
- Order affects the final image.
- Use clear notation and sketches to show each stage.
- Example: Reflect a shape across a line, then rotate it 90° anticlockwise around a point.
At-a-glance summary
Comparison table
| Transformation | Step | Detail | Example for a Triangle ABC with vertices A(1,2), B(3,2), C(2,4) |
|---|---|---|---|
| Reflection | Identify line of reflection | Line y=2 | Reflect ABC across y=2: A(1,2) stays, B(3,2) stays, C(2,4) maps to C'(2,0) |
| Rotation | Identify centre and angle | Centre O(0,0), rotate 90° anticlockwise | Rotate ABC: A(1,2) → A'(-2,1), B(3,2) → B'(-2,3), C(2,4) → C'(-4,2) |
| Enlargement | Identify centre and scale factor | Centre O(0,0), scale factor 2 | Enlarge ABC: A(1,2) → A'(2,4), B(3,2) → B'(6,4), C(2,4) → C'(4,8) |
| Translation | Identify vector | Vector (2, -1) | Translate ABC: A(1,2) → A'(3,1), B(3,2) → B'(5,1), C(2,4) → C'(4,3) |
Active recall
Quick check
1. What happens to the orientation of a shape after reflection?
The orientation reverses; the shape becomes a mirror image across the line of reflection.
2. If a shape is rotated 180° about a point, what changes occur?
The shape keeps its size and shape but is turned upside down (rotated), changing its orientation.
3. How does a scale factor of 0.5 affect a shape during enlargement?
The shape is reduced to half its size, maintaining shape and orientation.
4. Describe the effect of translating a shape by vector (4, -3).
Every point of the shape moves 4 units right and 3 units down, with no change in size or orientation.