Translations

Intuition and core definition

A translation slides every point by the same vector. It preserves distances, angles, parallelism, orientation, and shape, so it is a rigid motion. No nonzero translation has a fixed point in the plane.

Notation, language, and conditions

Vector $\langle a,b\rangle$ gives horizontal and vertical displacement. Coordinate rule $(x,y)\mapsto(x+a,y+b)$. The vector from original $P$ to image $P\prime$ must be identical for every point.

Why this idea matters

A translation adds one vector to every point, preserving size, orientation, and parallel direction without creating a fixed centre.

A dependable method

1. Read the horizontal and vertical components with signs.
2. Add $a$ to each $x$-coordinate and $b$ to each $y$-coordinate.
3. Keep vertex labels in corresponding order.
4. Draw parallel equal-length arrows from originals to images.
5. Subtract coordinates to verify every displacement vector matches.

Worked example

Common mistakes