What is a transformation? In geometry, a transformation is a way of moving or changing a geometric shape. The original shape is called the object, and the new shape after the movement is called the image.
Translation: A translation means sliding a shape across a flat surface without turning it, flipping it, or changing its size. Every point of the shape moves by the exact same distance and in the exact same direction. Think of pushing a chess piece forward on a chessboard, or sliding a video game character across the screen. We can describe a translation using a column vector:
Where \(x\) tells us how many steps to move horizontally (right is positive, left is negative) and \(y\) tells us how many steps to move vertically (up is positive, down is negative).
Reflection: A reflection is a flip over a straight line called the line of reflection (or mirror line). The image looks like a mirror reflection of the object. Think of looking at a mountain reflected in a perfectly still lake. Every single point on the image is at the exact same distance from the mirror line as its corresponding point on the original object, but on the completely opposite side.
Rotation: A rotation means turning a shape around a fixed point called the center of rotation. To perfectly describe a rotation, you need three specific details:
- The center of rotation (the point you pin down with your finger while turning).
- The angle of rotation (how far you turn, such as \(90^\circ\) or \(180^\circ\)).
- The direction of rotation (clockwise or counter-clockwise).
Think of the moving hands on an analog clock or turning a steering wheel.
| Transformation Type | Common Action Word | What Changes? | What Stays the Same? |
|---|---|---|---|
| Translation | Slide | Position only | Size, Shape, Orientation |
| Reflection | Flip | Position and Orientation | Size, Shape |
| Rotation | Turn | Position and Orientation | Size, Shape |