Transformational Geometry and Symmetry
Transformational geometry is the study of how shapes can be moved, flipped, turned, or resized while maintaining certain properties. These movements are called transformations, and they help us understand the relationships between different geometric figures.
The Four Basic Transformations
- Translation (Slide): Moving a figure in a straight line without rotating, reflecting, or changing its size or shape. Every point of the figure moves the same distance and in the same direction.
- Rotation (Turn): Turning a figure around a fixed point called the center of rotation. The figure maintains its size and shape.
- Reflection (Flip): Creating a mirror image of a figure across a line called the line of reflection. The original figure and its reflection are the same distance from the line but on opposite sides.
- Dilation (Resize): Enlarging or reducing a figure by a scale factor from a fixed point called the center of dilation. The figure changes in size but maintains its shape and proportions.
Coordinate Notation
When working with transformations on a coordinate plane, we can describe them using specific notation:
- Translation: (x, y) → (x + h, y + k), where h is the horizontal shift and k is the vertical shift
- Rotation: For common rotations around the origin:
- 90° counterclockwise: (x, y) → (-y, x)
- 180° counterclockwise: (x, y) → (-x, -y)
- 270° counterclockwise: (x, y) → (y, -x)
Visual Examples of Rotations (counterclockwise): The blue figure is the original, the orange figure is after rotation. The black dot marks the center of rotation.