Position and Transformations

An enlargement is the one transformation that changes a shape's size. It needs two pieces of information: a scale factor and a centre of enlargement.

The scale factor tells you how many times longer every length becomes, and the centre is the fixed point the shape grows away from. To enlarge, measure the distance from the centre to each corner, multiply by the scale factor, and plot the new corner.

Two facts to keep in mind. The angles stay the same, so the image is the same shape — the shapes are similar. But the area scales by the factor squared: with scale factor $3$, the area becomes $9$ times bigger.

This year you push the idea further. A scale factor does not have to be a whole number bigger than $1$ — it can be a fraction, or even a negative number, and each behaves in its own interesting way.

A scale factor does not have to make a shape bigger. A scale factor between $0$ and $1$ — a proper fraction — makes the shape smaller.

• Scale factor $\frac{1}{2}$ — every length halves.
• Scale factor $\frac{1}{3}$ — every length is one third as long.

It is still called an "enlargement", even though the image shrinks — the word covers any change of size.

The method is exactly the same as before: measure each corner's distance from the centre, multiply by the fraction, and plot the new corner closer in.

The area rule still uses the square. With scale factor $\frac{1}{2}$ the area becomes $\left(\frac{1}{2}\right)^2 = \frac{1}{4}$ of the original — a quarter, not a half. A small change in length makes a bigger change in area.

A scale factor can also be negative. A negative scale factor sends the image to the opposite side of the centre of enlargement, and turns it upside down.

To enlarge by a negative scale factor, measure from the centre to a corner as usual, then plot the new corner the matching distance on the other side of the centre.

The size still depends on the size of the number, ignoring the sign:

• Scale factor $-1$ — same size, but flipped through the centre.
• Scale factor $-2$ — twice as big, and flipped through the centre.
• Scale factor $-\frac{1}{2}$ — half the size, and flipped through the centre.

So a negative scale factor does two jobs at once: it changes the size by the size of the number, and it rotates the shape $180\degree$ about the centre. Take care to plot every corner on the far side.

A translation slides a shape, and it is written as a column vector — two numbers stacked in a bracket: $\begin{pmatrix} x \\ y \end{pmatrix}$ The top number is the move left or right (right is positive); the bottom number is the move up or down (up is positive).

This year you do two new things with vectors. First, you add them. If a shape is translated by $\begin{pmatrix} 4 \\ 1 \end{pmatrix}$ and then by $\begin{pmatrix} 2 \\ -3 \end{pmatrix}$, the single combined slide is the sum: $\begin{pmatrix} 4 \\ 1 \end{pmatrix} + \begin{pmatrix} 2 \\ -3 \end{pmatrix} = \begin{pmatrix} 6 \\ -2 \end{pmatrix}.$ Add the tops, add the bottoms.

Second, you find the vector between two points by subtracting the coordinates: the vector to point $B$ from point $A$ is $B - A$. The vector from $(1, 2)$ to $(5, 7)$ is $\begin{pmatrix} 5 - 1 \\ 7 - 2 \end{pmatrix} = \begin{pmatrix} 4 \\ 5 \end{pmatrix}$.

When a shape is transformed twice — one transformation, then another — the result is a combined transformation. The Stage 9 skill is to describe the single transformation that gives the same overall result.

Some neat results are worth knowing:

• Two reflections in parallel mirror lines combine to make a single translation.
• Two reflections in mirror lines that cross combine to make a single rotation.
• Two rotations about the same centre combine into one rotation — add the angles.
• Two translations combine into one translation — add the column vectors.

The order can matter. Reflect-then-rotate may land somewhere different from rotate-then-reflect, so always carry out the steps in the order given. When you have finished, compare the first shape with the last and ask: what single transformation gets me straight there? Describe it fully — name the mirror line, or give the centre, angle and direction, or state the column vector.

Getting confident with position and transformations sets you up well for what comes next:

• Similar shapes — fractional and negative enlargements lead straight into similarity, scale drawings and maps.
• Vectors — column vectors here are your first proper step into the wider study of vectors.
• Graphs and functions — translating and reflecting graphs uses exactly these ideas.
• In everyday life, transformations appear in computer graphics, animation, design, photography and architecture.

If a later geometry or graphing topic feels tricky, it is often a transformation or vector skill underneath. Keep this guide handy — describing and combining transformations gets quicker every time you practise.