Transformations – Moving and Resizing Shapes
A geometric transformation is a rule that moves, flips, turns, or resizes a shape. The original shape is called the object and the result is the image. Transformations are used in computer graphics, design, robotics, and map projections.
The Four Main Transformations
| Transformation | What It Does | Preserves Size? | Preserves Shape? |
|---|---|---|---|
| Translation | Slides the shape in a given direction | Yes | Yes |
| Rotation | Turns the shape around a fixed point | Yes | Yes |
| Reflection | Flips the shape over a mirror line | Yes | Yes |
| Enlargement | Scales the shape by a scale factor | No | Yes |
Translation
A translation moves every point the same distance in the same direction. It is described by a column vector (a over b), meaning move a units right (negative for left) and b units up (negative for down).
New position: (3 + 4, 2 + (−1)) = (7, 1).
Rotation
A rotation turns a shape about a fixed centre of rotation by a given angle (clockwise or anti-clockwise). You must specify: angle, direction, and centre.
Rule for 90° anti-clockwise: (x, y) → (−y, x). So (3, 1) → (−1, 3).
Rotation Rules About the Origin
| Rotation | Transformation Rule |
|---|---|
| 90° clockwise | (x, y) → (y, −x) |
| 90° anti-clockwise | (x, y) → (−y, x) |
| 180° (either direction) | (x, y) → (−x, −y) |
Reflection
A reflection flips a shape over a mirror line. Every point maps to its mirror image, equidistant from the line on the other side.
| Mirror Line | 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) |
Enlargement
An enlargement scales a shape by a scale factor from a centre of enlargement. Scale factor k > 1 makes it bigger; 0 < k < 1 makes it smaller; negative k flips and scales it.
Multiply each coordinate by 2: (2, 2), (6, 2), (2, 6). The image is twice the size.
Key Takeaways
- Translation: describe with a column vector. Image is congruent to object.
- Rotation: specify angle, direction, and centre of rotation.
- Reflection: specify the mirror line. Image is congruent, but orientation reverses.
- Enlargement: specify scale factor and centre. Image is similar (not congruent unless k = 1 or −1).
Practice Questions
- Translate point (4, −2) by vector (−3 over 5). State the image coordinates.
- Rotate point (5, 2) by 90° clockwise about the origin. State the image.
- Reflect point (3, 4) in the y-axis. State the image.
- Reflect (2, 5) in the line y = x. State the image.
- A triangle has vertices (1, 2), (3, 2), (2, 4). It is enlarged by scale factor 3 from the origin. Find the image vertices.
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