Parallel Translation (Translation)
Big Idea: A parallel translation (or simply a translation) is a geometric transformation that slides every point of a figure the same distance in the same direction — without rotating, reflecting, or resizing it.
Formal Definition
A translation is a transformation $T_{\vec{v}}$ that maps every point $P(x, y)$ to a new point $P'(x', y')$ such that:
$$P'(x', y') = (x + a,\; y + b)$$
where $\vec{v} = \langle a, b \rangle$ is the translation vector that defines the direction and distance of the shift.
- $a$ = horizontal shift (positive → right, negative → left)
- $b$ = vertical shift (positive → up, negative → down)
- $|\vec{v}| = \sqrt{a^2 + b^2}$ = distance each point moves
Key Properties
| Property | Preserved? |
|---|---|
| Distances (side lengths) | ✅ Yes |
| Angle measures | ✅ Yes |
| Parallelism of lines | ✅ Yes |
| Orientation (order of vertices) | ✅ Yes |
| Position | ❌ No (it changes!) |
Because distances and angles are preserved, a translation is an isometry — the image is always congruent to the original.