Translation

Geometry
definition

Also known as: slide, shift, sliding motion

Grade 9-12

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A rigid transformation that slides every point of a figure the same distance in the same direction. The simplest rigid motion; understanding translation builds the foundation for vectors and transformations.

This concept is covered in depth in our Geometry Transformations Guide, with worked examples, practice problems, and common mistakes.

Definition

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

💡 Intuition

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

🎯 Core Idea

Translation is a rigid motion that preserves size, shape, and orientation—only position changes.

Example

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

Formula

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

Notation

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

🌟 Why It Matters

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

💭 Hint When Stuck

Pick any vertex and move it by the translation vector. Then move every other vertex by the exact same amount and 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

🚧 Common Stuck Point

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

⚠️ 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

Frequently Asked Questions

What is Translation in Math?

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

Why is Translation important?

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

What do students usually 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 Translation?

Before studying Translation, you should understand: transformation geo.

Prerequisites

transformation geo

How Translation Connects to Other Ideas

To understand translation, you should first be comfortable with transformation geo. Once you have a solid grasp of translation, you can move on to vector intuition and composition of transformations.

Want the Full Guide?

This concept is explained step by step in our complete guide:

Geometry Transformations and Cross-Sections Guide →
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Interactive Playground

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