what is translation in geometry

In geometry, a translation is a type of transformation that slides a shape, point, or figure from one position to another without rotating, flipping, stretching, or resizing it.

Basic idea

A translation moves every point of a figure by the same distance and in the same direction.

Because the shape doesn’t turn or change size, the original figure and its image are congruent (same size and shape but in a different location).

Think of it like sliding a book across a table: it stays the same, but its position changes.

How translation works on a coordinate plane

On the coordinate plane, a translation is often described by a vector such as (a,b)(a,b)(a,b), where:

• aaa tells how far to move left or right (horizontal),
• bbb tells how far to move up or down (vertical).

For any point (x,y)(x,y)(x,y), the translated point becomes:

(x+a,y+b)(x+a,y+b)(x+a,y+b)

This rule is applied to all points of the figure.

Properties of a translation

• Preserves:
  • Lengths of segments,
  • Angles,
  • Shape and size (rigid transformation).

• Does not preserve:
  • The exact position in the plane (since the figure has moved).

Translations are one of the four basic geometric transformations , along with reflection, rotation, and dilation.

Example (simple)

Suppose a triangle has vertices at:

• A(1,1)A(1,1)A(1,1),
• B(3,1)B(3,1)B(3,1),
• C(2,3)C(2,3)C(2,3).

If you translate it by (2,−1)(2,-1)(2,−1), each point moves:

• A(1,1)→(1+2,1−1)=(3,0)A(1,1)→(1+2,1-1)=(3,0)A(1,1)→(1+2,1−1)=(3,0),
• B(3,1)→(5,0)B(3,1)→(5,0)B(3,1)→(5,0),
• C(2,3)→(4,2)C(2,3)→(4,2)C(2,3)→(4,2).

The new triangle has the same shape and size but is shifted right by 2 units and down by 1 unit.

Quick comparison of basic transformations

Transformation| What it does| Changes size?| Changes orientation?
---|---|---|---
Translation| Slides figure without turning or resizing 13| No 37| No 3
Reflection| Mirrors figure over a line 58| No 5| Yes (image is flipped) 8
Rotation| Turns figure around a point 58| No 5| Yes (rotated image) 8
Dilation| Stretches or shrinks figure 58| Yes 5| No (just size) 8

Information gathered from public educational resources and portrayed here.