Chapter overview



The four transformations

Your lesson progress



The four transformations

In a nutshell

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


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

Example 1

The square ABCDABCD 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.

Maths; Shapes and area; KS4 Year 10; The four transformations

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

The corresponding column vector is therefore (55)\begin{pmatrix}-5\\-5\end{pmatrix} as AA moved 55 steps to the left and 55 steps down to end up at (4,4)(-4,-4).

The transformation is a translation by column vector (55)\underline{\begin{pmatrix}-5\\-5\end{pmatrix}}.


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 ABCDABCD to the shape ABCDA'B'C'D' in the diagram below.

Maths; Shapes and area; KS4 Year 10; The four transformations

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

Note: In this case, there was no need to specify direction as a rotation of 180180^\circ clockwise is the same as a rotation of 180180^\circ