Chapter overview Maths

Exam board

AQA

Number

Algebra

Graphs

Ratio proportion and rates of change

Shapes and area

Angles and geometry

Trigonometry

Probability

Statistics

Maths

# The four transformations  0%

Summary

# The four transformations

## In a nutshell

There are four main ways to mathematically alter a shape: translation, rotation, reflection and enlargement.

## Translation

Translating a shape means to move its position while keeping the orientation and size of the shape intact. Translations are described using column vectors $\begin{pmatrix}x\\y\end{pmatrix}$​, which means to move the shape $x$​ spaces to the right and $y$​ spaces up.

##### Example 1

The square $ABCD$ was originally in the top-right quadrant. It was then transformed to the bottom-left quadrant as shown below. Describe the transformation that took place.

The size and orientation of the shape is unchanged, so the shape has been translated.

To find the corresponding column vector, focus on one vertex and see where it moved to.

$A=(1,1)$ gets mapped to $A=(-4,-4)$.

The corresponding column vector is therefore $\begin{pmatrix}-5\\-5\end{pmatrix}$ as $A$ moved $5$ steps to the left and $5$ steps down to end up at $(-4,-4)$.

The transformation is a translation by column vector $\underline{\begin{pmatrix}-5\\-5\end{pmatrix}}$.

## Rotation

Rotating a shape has three details associated with it:

1. The angle by which the shape has been rotated.
2. The direction of rotation (clockwise or anticlockwise).
3. The centre of rotation.

The centre of rotation is the point you rotate everything around.

##### Example 2

Describe the transformation that maps the shape $ABCD$ to the shape $A'B'C'D'$ in the diagram below.

This is a rotation of $\underline{180 \degree}$ with centre O.

Note: In this case, there was no need to specify direction as a rotation of $180^\circ$ clockwise is the same as a rotation of $180^\circ$