Translation | Moosa Academy

A translation slides a figure from one place to another without turning or flipping it. Every point moves the same distance in the same direction, so the figure keeps its shape, its size, and the way it faces.

Theorem Translation by a vector

A translation is fixed by one vector: every point is moved the same displacement, so $(x,y)$ maps to $(x+a,\;y+b)$ .

Sliding by the vector $(a,b)$ :

(x,\,y) \longrightarrow (x+a,\;y+b)

Note What a translation keeps

A translation preserves distances and angles, so the image is congruent to the original. Unlike a reflection, it also keeps the orientation — the figure faces the same way and is not flipped.

Every point actually moves (as long as the vector is not zero), so a non-trivial translation has no fixed points.

Example Translating a point

Translate the point $P = (1,2)$ by the vector $(4,-1)$ .

(x,\,y) \longrightarrow (x+4,\;y-1)

P = (1,\,2) \longrightarrow P' = (1+4,\;2-1)

⟹ P′ = (5, 1).

Example Translating a triangle

Translate triangle with vertices $A(0,0)$ , $B(2,0)$ , $C(0,3)$ by the vector $(3,1)$ . Move each vertex by the same rule.

A(0,0) \longrightarrow A'(3,1)

B(2,0) \longrightarrow B'(5,1)

C(0,3) \longrightarrow C'(3,4)

⟹ △A′B′C′ ≅ △ABC, shifted and facing the same way.

Summary

A translation slides every point the same distance in the same direction, set by one vector.
The coordinate rule is $(x,y) \to (x+a,\;y+b)$ .
It preserves distance, angle, and orientation — the image is congruent and faces the same way.
A non-zero translation moves every point, so it has no fixed points.