geometric transformations and Congruence
Geometric transformations are functions that map one geometric figure to another according to specific rules. In grade 10 geometry, we focus on rigid transformations (also called isometries), which preserve the size and shape of figures—only their position or orientation changes.
The Four Rigid Transformations
- Translations: Sliding a figure along a straight line without rotating or flipping it. A translation moves every point of a figure the same distance in the same direction.
- Reflections: Flipping a figure across a line (the line of reflection) to create a mirror image.
- Rotations: Turning a figure around a fixed point (the center of rotation) by a specified angle.
- Glide Reflections: A composition of a reflection and a translation parallel to the line of reflection.
Mathematical Representation
We can represent transformations both geometrically and algebraically:
- Geometric representation: Visual depiction of how points move from their original positions to new positions.
- Algebraic representation: Mathematical formulas that describe how coordinates change under each transformation.
| Transformation | Algebraic Representation |
|---|---|
| Translation by vector (h, k) | (x, y) → (x + h, y + k) |
| Reflection across x-axis | (x, y) → (x, -y) |
| Reflection across y-axis | (x, y) → (-x, y) |
| Reflection across y = x | (x, y) → (y, x) |
| Rotation of 90° counterclockwise around origin | (x, y) → (-y, x) |
| Rotation of 180° around origin | (x, y) → (-x, -y) |
| Rotation of 270° counterclockwise around origin | (x, y) → (y, -x) |