Transformations involve changing the position, size, or orientation of shapes on a 2D plane. Here's a summary of key concepts:
1. Congruence
- Definition: Two shapes are congruent if they have the same size and shape.
- Congruence remains true even after rotation or reflection.
- Key Fact: Congruence is comparative; a shape cannot be congruent on its own without a reference.
- Example: Shapes A, B, C, and D (rotated and reflected versions of the same shape) are congruent.
2. Symmetry
Line Symmetry
- Definition: A shape is symmetrical if it can be divided into two identical halves by a line of symmetry.
- Key Fact: Regular polygons have as many lines of symmetry as their number of sides (e.g., a hexagon has 6). A circle has infinite symmetry.
Rotational Symmetry
- Definition: A shape has rotational symmetry if it looks the same after a rotation about a point.
- Order of Rotational Symmetry: The number of times the shape matches itself in one full rotation.
- Example: The recycling symbol has an order of 3.
3. Reflection
- Definition: A shape is flipped across a line (the mirror line) to produce a reflection.
- Key Rules:
- Points and their reflections are equidistant from the mirror line.
- The line joining a point and its reflection is perpendicular to the mirror line.
- Drawing Reflections: Use graph paper and measure accurately to create reflected shapes.
4. Translation
- Definition: Moving a shape in a straight line without rotating or flipping it.
- Key Concept: Every point in the shape moves the same distance and direction.
- Vectors:
- Translation is represented as a column vector ( \begin{bmatrix} x \ y \end{bmatrix} ), where ( x ) is the horizontal movement and ( y ) is the vertical movement.
- Example: ( \begin{bmatrix} 5 \ -2 \end{bmatrix} ) means moving 5 units right and 2 units down.
5. Rotation
- Definition: Turning a shape around a fixed point (centre of rotation) through a specific angle (clockwise or anticlockwise).
- Key Rule: The distance of any point on the shape from the centre of rotation remains constant.
- Example:
- Rotate a triangle 90° anticlockwise or 180° clockwise around a point.
- Use a compass to trace the rotation path (circle).
Practical Applications?
These transformations are foundational in:
- Design and Architecture: Creating symmetrical or rotated patterns.
- Everyday Tasks: Matching wallpaper patterns or arranging tiles.
- Engineering and Physics: Using vectors for displacement and forces.
By mastering these basic transformations, you'll develop spatial reasoning and enhance your problem-solving skills!