Translations

What are transformations in maths?

There are four transformations to learn
- translations, rotations, reflections and enlargements
A transformation can change the position, orientation and/or size of a shape
- The original shape is called the object
- The transformed shape is called the image
Vertices are labelled to show corresponding points
- Vertices on the object are labelled A, B, C, etc.
- Vertices on the image are labelled A’, B’, C’ etc.

What is a translation?

A translation moves a shape
The size and orientation (which way up it is) of the shape stays the same
- The object and image are congruent

What is a translation vector?

The movement of a translation is described by a vector
You need to know how to write a translation using a vector (rather than words)
Vectors are written as column vectors in the form where:
- is the distance moved horizontally
  - Negative means move to the left
  - Positive means move to the right
- is the distance moved vertically
  - Negative means move down
  - Positive means move up

How do I translate a shape?

STEP 1
Interpret the translation vector
- $(\begin{matrix} 3 \\ - 1 \end{matrix})$ means 3 to the right and 1 down
STEP 2
Move each vertex on the original object according to the vector
STEP 3
Connect the new vertices and label the translated image
- It should look identical to the original object just in a different position
In some cases the image can overlap the object

How do I describe a translation?

To describe a translation, you must:
- State that the transformation is a translation
- Give the column vector that describes the movement
To find the vector:
- Pick a point on the original shape
- Identify the corresponding point on the image
- Count how far left or right () you need to go from the object to get to the image
  - If you go to the left then $x$ will be a negative number
- Count how far up or down () you need to go from the object to get to the image
  - If you go down then $y$ will be a negative number
- Write these numbers as a vector
  - $(\begin{matrix} x \\ y \end{matrix})$

Exam Tip

The vector is how the shape moves not the size of the gap between the object and the image
- Watch out for this common error!
Use tracing paper to check your answer

Worked example

(a)

\begin{matrix}  \end{matrix}

On the grid below translate shape P using the vector

(\begin{matrix} - 4 \\ 5 \end{matrix})

Label your translated shape P'.

Grid showing an object P

The vector means "4 to the left" and "5 up"
You don't have to draw in any arrows but it is a good idea to mark your paper after counting across and up a couple of times to check that you are in the correct place

A grid showing the translation of a vertex on an object P

Translating one vertex and then following around the shape one vertex at a time makes it easier to get the shape in exactly the right position!

A grid showing an object P and its translated image P'

(b)

Describe fully the single transformation that creates shape B from shape A.

A grid showing an object A and its transformed image B

This is a case where the image overlaps the object
You should still see that the shape is the same size and the same way up so it is a translation
Start at a vertex on the original object that is well away from any overlap area to avoid confusion and count the number of position left/right and up/down that you need to move to reach the corresponding vertex on the translated image
Take care when counting around the axes!

A grid showing the translation of a vertex between an object A and its image B

Shape A has been translated using the vector $(\begin{matrix} 2 \\ - 3 \end{matrix})$

Translations (CIE IGCSE Maths: Core)

Revision Note

Translations

What are transformations in maths?

What is a translation?

What is a translation vector?

How do I translate a shape?

How do I describe a translation?

Exam Tip

Worked example

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Author: Dan