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)(x,y)\mapsto(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)(x+3, y+2).

Notation

aa is horizontal shift and bb 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→RnT_{\vec{v}}: \mathbb{R}^n \to \mathbb{R}^n defined by Tv⃗(P)=P+v⃗T_{\vec{v}}(P) = P + \vec{v}; Tv⃗T_{\vec{v}} is an isometry: ∣Tv⃗(P)−Tv⃗(Q)∣=∣P−Q∣  ∀P,Q|T_{\vec{v}}(P) - T_{\vec{v}}(Q)| = |P - Q|\;\forall P, Q

Worked Examples

Example 1

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

Answer

A′=(7,1)A' = (7, 1)

First step

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

Full solution

  1. 2
    Step 2: A(2,−3)A(2, -3) translated by (5,4)(5, 4): (2+5, −3+4)=(7,1)(2+5,\ -3+4) = (7, 1).
  2. 3
    Step 3: A′=(7,1)A' = (7, 1).
A translation slides every point by the same amount in the same direction. The translation vector (5,4)(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)A(0,0), B(3,0)B(3,0), C(0,4)C(0,4) is translated by (−2,3)(-2, 3). Find the new vertices and confirm the side lengths are unchanged.

Example 3

medium
Triangle with vertices A(1,2)A(1,2), B(4,2)B(4,2), C(3,5)C(3,5) is translated by ⟨−3,4⟩\langle -3, 4 \rangle. 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?

aa is horizontal shift and bb 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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