Translation Formula

Translation is a rigid transformation that slides every point of a figure the same distance in the same direction.

The Formula

(x,y)↦(x+a,y+b)

When to use: Sliding a chess piece straight across the board—every point moves the same amount, same direction.

Quick Example

Translate triangle 3 units right and 2 up: every point moves (x+3,y+2).

Notation

a is horizontal shift and b is vertical shift.

What This Formula Means

A rigid transformation that slides every point of a figure the same distance in the same direction.

Sliding a chess piece straight across the board—every point moves the same amount, same direction.

Formal View

Tv⃗:Rn→Rn defined by Tv⃗(P)=P+v⃗; Tv⃗ is an isometry: ∣Tv⃗(P)−Tv⃗(Q)∣=∣P−Q∣  ∀P,Q

Worked Examples

Example 1

easy
Translate the point A(2,−3) by the vector (5,4). Where does A end up?

Answer

A′=(7,1)

First step

1
Step 1: A translation by (a,b) maps (x,y)→(x+a,y+b).

Full solution

  1. 2
    Step 2: A(2,−3) translated by (5,4): (2+5, −3+4)=(7,1).
  2. 3
    Step 3: A′=(7,1).
A translation slides every point by the same amount in the same direction. The translation vector (5,4) means move 5 units right and 4 units up. Translations preserve distances, angles, and orientation — they are rigid motions.

Example 2

medium
Triangle with vertices A(0,0), B(3,0), C(0,4) is translated by (−2,3). Find the new vertices and confirm the side lengths are unchanged.

Example 3

medium
Triangle with vertices A(1,2), B(4,2), C(3,5) is translated by ⟨−3,4⟩. Find the new vertices.

Common Mistakes

  • Changing only one coordinate when the move has horizontal and vertical parts — apply both parts to every point.
  • Calling any movement a translation — rotations and reflections move points differently.
  • Changing size during a slide — translations preserve length and angle measures.

Why This Formula Matters

Translations build coordinate-rule fluency and help students understand congruence as motion-preserved shape. Recognizing it by "Did every point move by the same vector?" — rather than by familiar numbers — is what lets a student tell it apart from rotation and reflection in a mixed problem set.

Frequently Asked Questions

What is the Translation formula?

A rigid transformation that slides every point of a figure the same distance in the same direction.

How do you use the Translation formula?

Sliding a chess piece straight across the board—every point moves the same amount, same direction.

What do the symbols mean in the Translation formula?

a is horizontal shift and b is vertical shift.

Why is the Translation formula important in Math?

Translations build coordinate-rule fluency and help students understand congruence as motion-preserved shape. Recognizing it by "Did every point move by the same vector?" — rather than by familiar numbers — is what lets a student tell it apart from rotation and reflection in a mixed problem set.

What do students get wrong about Translation?

The procedure for translation is the easy part; the trap is changing only one coordinate when the move has horizontal and vertical parts. Asking "Did every point move by the same vector?" first is what keeps a correct-looking calculation from being attached to the wrong concept.

What should I learn before the Translation formula?

Before studying the Translation formula, you should understand: transformation geo.

Want the Full Guide?

This formula is covered in depth in our complete guide:

Geometry Transformations and Cross-Sections Guide →
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