Translation Formula

The Formula

(x, y) \to (x + a, y + b) where (a, b) is the translation vector

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

T_{(a,b)} denotes a translation by vector (a, b)

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

T_{\vec{v}}: \mathbb{R}^n \to \mathbb{R}^n defined by T_{\vec{v}}(P) = P + \vec{v}; T_{\vec{v}} is an isometry: |T_{\vec{v}}(P) - T_{\vec{v}}(Q)| = |P - Q|\;\forall P, Q

Worked Examples

Example 1

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

Solution

  1. 1
    Step 1: A translation by (a, b) maps (x, y) \to (x+a, y+b).
  2. 2
    Step 2: A(2, -3) translated by (5, 4): (2+5,\ -3+4) = (7, 1).
  3. 3
    Step 3: A' = (7, 1).

Answer

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 \langle -3, 4 \rangle. Find the new vertices.

Common Mistakes

  • Moving different points by different amounts — in a translation every point moves the same distance in the same direction
  • Confusing translation with rotation — translation slides without turning
  • Applying the translation vector with the wrong sign — translating right by 3 means adding 3 to x, not subtracting

Why This Formula Matters

The simplest rigid motion; understanding translation builds the foundation for vectors and transformations.

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?

T_{(a,b)} denotes a translation by vector (a, b)

Why is the Translation formula important in Math?

The simplest rigid motion; understanding translation builds the foundation for vectors and transformations.

What do students get wrong about Translation?

Every single point moves exactly the same distance in exactly the same direction—no rotation occurs.

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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