Changes and invariance under transformations
The four standard transformations all change a shape's position. Some preserve more than others. Understanding invariance — what stays the same — is essential.
The four transformations
| Transformation | Defined by | Preserves |
|---|---|---|
| Translation | Vector (x, y) | Size, shape, orientation |
| Reflection | Mirror line | Size, shape (reverses orientation) |
| Rotation | Centre + angle + direction | Size, shape, sense (if angle 0/180°) |
| Enlargement | Centre + scale factor | Shape only (size scales) |
Translations
Every point moves the same vector. Notation: T = (a, b) means right a, up b. Inverse: (−a, −b).
Reflections
A point and its image are equidistant from the mirror line, on opposite sides. Common mirrors: x-axis, y-axis, y = x, y = −x.
| Mirror | Coordinate rule |
|---|---|
| x-axis (y = 0) | (x, y) → (x, −y) |
| y-axis (x = 0) | (x, y) → (−x, y) |
| y = x | (x, y) → (y, x) |
| y = −x | (x, y) → (−y, −x) |
Rotations
Rotate by an angle (commonly 90°, 180°, 270°) about a centre.
| Rotation | Rule (about origin) |
|---|---|
| 90° clockwise | (x, y) → (y, −x) |
| 90° anticlockwise | (x, y) → (−y, x) |
| 180° | (x, y) → (−x, −y) |
Enlargement (G7)
Centre + SF. Shape preserved; sizes scale by SF.
Combining transformations
If you apply two transformations in sequence, the result is sometimes equivalent to a single transformation.
Examples:
- Two reflections in parallel mirrors = a translation.
- Two reflections in intersecting mirrors = a rotation about their intersection.
- A rotation + a translation = generally an isometry.
Invariance — what stays the same
| Property | Trans | Refl | Rot | Enl |
|---|---|---|---|---|
| Length | ✓ | ✓ | ✓ | × |
| Angles | ✓ | ✓ | ✓ | ✓ |
| Area | ✓ | ✓ | ✓ | × (× SF²) |
| Orientation | ✓ | × (flipped) | ✓ | ✓ (or flipped if SF < 0) |
| Parallel lines stay parallel | ✓ | ✓ | ✓ | ✓ |
⚠Common mistakes
- Confusing rotation direction — clockwise vs anticlockwise.
- Forgetting to fully describe — translations need a vector; reflections a mirror line; rotations centre + angle + direction; enlargements centre + SF.
- Mixing up order of transformations — A then B is generally NOT the same as B then A.
- Saying "the angle of rotation is 90°" without specifying clockwise/anticlockwise.
- Not preserving orientation under enlargement — a positive SF preserves; negative inverts.
➜Try this— Quick check
A point (3, 4) is reflected in the y-axis, then translated by (2, 1). Find the final image.
- After reflection: (−3, 4).
- After translation: (−1, 5).
AI-generated · claude-opus-4-7 · v3-deep-geometry