Transformations: Translation, Reflection, Rotation
Geometry Module 2: Geometric Transformations
Essential Understanding: Geometric transformations are operations that move or change a shape while preserving its essential properties. Master the three fundamental transformations to solve CSEC exam problems with confidence.
Core Concepts
Translation
A translation moves a shape without rotating or flipping it. Every point moves the same distance in the same direction.
Column Vector Notation:
\[ \begin{pmatrix} x \\ y \end{pmatrix} \rightarrow \begin{pmatrix} x+a \\ y+b \end{pmatrix} \]
where a = horizontal shift, b = vertical shift
Reflection
A reflection creates a mirror image across a line called the axis of reflection or mirror line. The shape appears "flipped."
Rotation
A rotation turns a shape around a fixed point called the center of rotation. The shape stays the same size but changes orientation.
Rotation Angle:
90°
Positive angles are anticlockwise
2. Interactive Transformation Lab
Drag the Blue Triangle (Object) to see the Red Triangle (Image) transform!
Interactive Rule Matrix
Your quick reference cheat sheet for coordinate transformations. These rules apply to any point (x, y):
| Transformation | Input Point | Mapping Rule | Output Point |
|---|---|---|---|
| ↗️ Translation | (x, y) | Add vector \[ \begin{pmatrix} a \\ b \end{pmatrix} \] | (x+a, y+b) |
| 🪞 Reflection (x-axis) | (x, y) | Flip y-coordinate | (x, -y) |
| 🪞 Reflection (y-axis) | (x, y) | Flip x-coordinate | (-x, y) |
| 🔄 Rotation 90° ACW | (x, y) | Cycle coordinates, negate new x | (-y, x) |
| 🔄 Rotation 180° | (x, y) | Negate both coordinates | (-x, -y) |
| 🔄 Rotation 90° CW | (x, y) | Cycle coordinates, negate new y | (y, -x) |
💡 Memory Tip
Reflection: Think of it as "flipping" the sign of the coordinate perpendicular to the mirror line.
Rotation 90°: Remember the pattern as a cycle: (x, y) → (-y, x) for anticlockwise, or (x, y) → (y, -x) for clockwise.
Step-by-Step Construction Guides
How to Translate: Vector Addition Method
Example: Translate by vector \[ \begin{pmatrix} 3 \\ -2 \end{pmatrix} \]
Triangle ABC with A(1, 2), B(4, 2), C(2, 5)
A'(1+3, 2+(-2)) = A'(4, 0)
B'(4+3, 2+(-2)) = B'(7, 0)
C'(2+3, 5+(-2)) = C'(5, 3)
How to Reflect: Distance to Mirror Line
How to Rotate: Center Point Method
Tracing Paper Method
For rotations, you can also use tracing paper:
- Trace the shape AND the center point on tracing paper
- Place a pin at the center point
- Rotate the paper exactly the required angle
- The new position shows your image!
CSEC Common Pitfalls
Direction Matters!
Always specify anticlockwise or clockwise when describing rotations. Positive angles are anticlockwise by default.
✓ "Rotate 90° anticlockwise about origin"
Mirror Lines: x vs y
x = k is vertical, y = k is horizontal. They're completely different!
y = 3: horizontal line through y = 3
Prime Notation
Always use the prime symbol (') to denote image points.
✓ Image: A', B', C'
Rotation Center
Never assume the center of rotation! It might be a vertex, midpoint, or other point — not necessarily the origin!