Translation in Geometry — Definition, Rules, Formula & Examples

Written by

Malcolm McKinsey

January 11, 2023

Fact-checked by

Paul Mazzola

Translation math definition

A translation in plane geometry is a type of geometric transformation where the original figure is moved in a given direction to new coordinates on the grid. In geometry, there are horizontal translations, vertical translations, or a combination of both.

Translation of shapes

Two-dimensional geometry translations allow you to move any shape or figure on the coordinate plane with great precision. When translating a shape, every vertex on the shape will move the same number of units.

A translation moves the shape from one location to another while the angles, lengths of sides, and total area of the shape remain the same.

[insert coordinate grid system with a simple shape appearing twice, with labels “Pre-image” and “Image” beneath each]

The properties of translations – that the image is a faithful reproduction of the original figure in every way (line length, angle, and size) except location – mean the image exactly matches the pre-image but appears in a new spot on the coordinate grid system.

Translation rules

Translating any shape or point on a coordinate plane follows certain guidelines:

1. Referring to the original shape or figure as the pre-image.

2. Referring to the newly located shape as the image.

3. Moving every part of the pre-image the same distance in the same direction to create the image

4. Allowing no rotation, expansion, dilation, shear, stretch, or rotation of the pre-image in placing the image

5. Using correct notation to indicate distance and direction of the translation

Translations are the simplest transformation in geometry and are often the first step in performing other transformations on a figure or shape.

For example, you may find you want to translate and rotate a shape. You can perform seven types of transformations on any shape or figure:

1. Translation – moving the shape without any other change

2. Rotation – turning the object around a given fixed point

3. Dilation – a decrease in scale (like a photocopy shrinkage)

4. Expansion – an increase in scale (like a photocopy enlargement)

5. Shear – a movement of all the shape’s points in one direction except for points on a given line (like a crate being collapsed)

6. Stretch – a one-way or two-way change using an invariant line and a scale factor (as if the shape were rubber)

7. Reflection – exchanging all points of a shape or figure with their mirror image across a given line (like looking in a mirror)

Translation notations

The original reference point for any figure or shape is presented with its coordinates, using the x-axis and y-axis system, (x,y).

Suppose you have Point P located at (3, 4). Point P the pre-image. To avoid confusion, the new image is indicated with a little prime stroke, like this: P′, and that point is pronounced “P prime.

How to do translations

You can know how to slide a shape using the ${T}_{(a,b)}$ system where T stands for translation, a stands for horizontal movement along the x-axis, and b stands for vertical movement up and down the y-axis.

Suppose you are given instructions for Point P at (3, 4) like this:

That system tells you Point P has a horizontal shift of 7 units and a vertical shift of 2 units, so we use this formula:

The new figure (the point) has a new x-coordinate of positive 10 and y-coordinate of 6.

That arrow, →, indicates the translation or slide. Translating Point P to Point P′ looks like this:

[insert drawing with Point P at (3,4) (original point) and Point P prime (new point); at (10,6)]

Translating a single point is not much fun. Let’s translate a line segment:

[insert professionally drawn coordinate grid with a line segment HI, Point H(-2,6) and Point I(1,2)]

We can write out our ${T}_{(a,b)}$ instruction so that we have to move the line to the left 1 units and down 2 units:

Now we know how to write the formula for each endpoint:

Start by translating point H:

Next, we translate point I:

The steps are the same for all geometric shapes with endpoints on the coordinate grid, whether you are translating a triangle, quadrilateral, or dodecagon. So long as you have your ${T}_{(a,b)}$ instructions, you know how to move every point.

Translation examples

Here is an irregular octagon translated following the instruction, ${T}_{(8,2)}$:

[insert professional version of this rough draft; maybe substitute the letters WHIZBANG for ABCDEFGH]

Here’s a septagon translated down 2 units and over to the right 4 units:

[Note: when professionally drawing from this rough draft, remove the vector ray HI please; consider using the letters BLAZING instead of ABCDEFG]

The ${T}_{(a,b)}$ notation for the septagon is ${T}_{(-2,4)}$.

Here’s a translation of a regular pentagon. Can you tell if it follows the notation ${T}_{(1,-6)}$ or ${T}_{(-6,-1)}$?

[please make a professional version of this rough draft; maybe make the letters MAGIC instead of ABCDE]

Let’s see. We moved Point A, at the lower left of our pentagon, from (4,2) to (-2, 1). We had to subtract 6 from 4 to get -2; we subtracted 1 from 2 to get 1, so we moved -6 and -1; the correct notation was ${T}_{(-6,-1)}$.

The translation works for any shape, even irregular shapes:

[please make a professional drawing with a cat shape but no vector, based on this rough draft]

Our mathematical cat has moved to the left and up, so the translation must include a negative x-value and a positive y-value.

Do you think it is ${T}_{(-10,3)}$ or ${T}_{(10,-3)}$ or ${T}_{(-3,10)}$? It is ${T}_{(-10,3)}$ because the first value is always the x-axis.