G7, G8, G24Geometry & MeasuresFoundation & HigherGrades 2–7

Transformations

A transformation moves or resizes a shape on a grid. There are four: translation (slide), reflection (flip), rotation (turn) and enlargement (resize). You need to perform each one accurately and — just as often on AQA papers — describe a transformation fully, which means giving every required detail.

50 min Video by Maths Genie AQA GCSE Maths

The big picture

Translations, reflections and rotations keep a shape the same size and shape, so the image is congruent to the original. Enlargements change the size but keep the shape, so the image is similar. That link to congruence and similarity is why transformations sit in the geometry strand.

Every description has a checklist. Translation: a column vector. Reflection: the mirror line. Rotation: the angle, the direction and the centre. Enlargement: the scale factor and the centre. Missing one item costs a mark.

Enlargement (Transformations of Shapes)Curated by Kepler, not produced by us. Watch it, then read the notes below.

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What you'll learn

Translate shapes using column vectors
Reflect shapes in horizontal, vertical and diagonal lines
Rotate shapes about a centre
Enlarge shapes by positive, fractional and negative scale factors, and describe transformations fully

Translations and reflections

A translation by $\begin{pmatrix} a \\ b \end{pmatrix}$ moves every point $a$ right (left if negative) and $b$ up (down if negative).

A reflection flips the shape in a mirror line. Each point of the image is the same perpendicular distance from the line as the original, on the other side. Common lines: $x = k$ (vertical), $y = k$ (horizontal), $y = x$ and $y = -x$ .

Triangle A reflected in the

y

-axis gives B; A reflected in the

x

-axis gives C.

Example 1The point

(3, -1)

is translated by

\begin{pmatrix} -5 \\ 4 \end{pmatrix}

. Where does it end up?

x

3 + (-5) = -2

y

-1 + 4 = 3

Answer

(-2, 3)

Your turnSame method, new numbers — try it before moving on.

The point $(-4, 2)$ is translated by $\begin{pmatrix} 6 \\ -7 \end{pmatrix}$ . Give the coordinates of its image.

Your answer

Type coordinates like (3, -1).

Show worked solution

Solution

1 $x$ : $-4 + 6 = 2$ ; $y$ : $2 - 7 = -5$ .

Answer: $(2, -5)$

Example 2Reflect the point

(5, 2)

in the line

y = x

1Reflecting in

y = x

swaps the coordinates.

Answer

(2, 5)

Your turnSame method, new numbers — try it before moving on.

Reflect the point $(1, 4)$ in the line $x = 3$ . Give the coordinates of its image.

Your answer

Type coordinates like (3, -1).

Show hint

The point is 2 units left of the line $x = 3$ .

Show worked solution

Solution

1 $(1, 4)$ is 2 units left of $x = 3$ .
2The image is 2 units right of it: $(5, 4)$ .

Answer: $(5, 4)$

Rotations

A rotation needs an angle, a direction (clockwise or anticlockwise) and a centre. Tracing paper is allowed in the exam: trace the shape, pin the centre with your pencil, and turn.

About the origin: $90^\circ$ clockwise sends $(x, y) \to (y, -x)$ ; $90^\circ$ anticlockwise sends $(x, y) \to (-y, x)$ ; $180^\circ$ sends $(x, y) \to (-x, -y)$ .

Example 1Rotate the point

(4, 1)

through

90^\circ

clockwise about the origin.

(x, y) \to (y, -x)

Answer

(1, -4)

Your turnSame method, new numbers — try it before moving on.

Rotate the point $(-3, 2)$ through $180^\circ$ about the origin. Give the image.

Your answer

Type coordinates like (3, -1).

Show worked solution

Solution

1 $180^\circ$ changes the sign of both coordinates: $(3, -2)$ .

Answer: $(3, -2)$

Tip — A $180^\circ$ rotation needs no direction — clockwise and anticlockwise give the same result.

Enlargements

An enlargement with scale factor $k$ and centre $C$ multiplies the distance of every point from $C$ by $k$ . Draw lines from the centre through each vertex to place the image.

A fractional scale factor such as $\frac{1}{2}$ makes the shape smaller (still called an enlargement). A negative scale factor puts the image on the opposite side of the centre, upside down.

Enlargement with scale factor 2, centre C: each vertex of A′ is twice as far from C as the matching vertex of A.

Example 1Enlarge the point

(5, 3)

by scale factor 3, centre

(2, 1)

1From the centre to the point:

\begin{pmatrix} 3 \\ 2 \end{pmatrix}

2Multiply by 3:

\begin{pmatrix} 9 \\ 6 \end{pmatrix}

3Add to the centre:

(2 + 9, 1 + 6)

Answer

(11, 7)

Your turnSame method, new numbers — try it before moving on.

Enlarge the point $(4, 6)$ by scale factor $-\frac{1}{2}$ , centre $(0, 2)$ . Give the image.

Your answer

Type coordinates like (3, -1).

Show hint

The vector from the centre is $\begin{pmatrix} 4 \\ 4 \end{pmatrix}$ . Multiply it by $-\frac{1}{2}$ .

Show worked solution

Solution

1Vector from centre: $\begin{pmatrix} 4 \\ 4 \end{pmatrix}$ .
2 $\times -\frac{1}{2}$ : $\begin{pmatrix} -2 \\ -2 \end{pmatrix}$ .
3Image: $(0 - 2, 2 - 2) = (-2, 0)$ .

Answer: $(-2, 0)$

Finding the centre

To find the centre of an enlargement from a diagram, draw straight lines through matching vertices of the object and image. They all meet at the centre. The scale factor is image length ÷ object length.

Describing and combining transformations

Describe with the single word and all its details: "Rotation, $90^\circ$ anticlockwise, centre $(1, 2)$ " or "Enlargement, scale factor $\frac{1}{3}$ , centre $(0, 0)$ ". Never describe a single transformation as a combination.

When two transformations are applied in turn, do them in order, then look for one transformation that has the same overall effect.

Example 1A shape is reflected in the

x

-axis and then in the

y

-axis. Describe the single transformation that has the same effect.

1Reflect in

x

-axis:

(x, y) \to (x, -y)

2Then in

y

-axis:

(x, -y) \to (-x, -y)

3That is the effect of a

180^\circ

rotation about the origin.

Answer

Rotation,

180^\circ

, centre

(0, 0)

Think like an examiner

Describe one transformation only. "Reflected then translated" scores zero when a single transformation is asked for.
Use the correct words: "translation", not "moved" or "slid"; give a column vector, not "3 right, 2 down".
Draw images accurately on the grid with a sharp pencil — vertices must be exact.

Describe fully

\text{Translation: } \begin{pmatrix} a \\ b \end{pmatrix}

The column vector.

\text{Reflection: mirror line}

Such as

y = -x

x = 2

\text{Rotation: angle, direction, centre; Enlargement: SF, centre}

Every detail earns a mark.

Watch out for these

Writing a column vector as

(3, -2)

in a description.

Use a column vector

\begin{pmatrix} 3 \\ -2 \end{pmatrix}

Reflecting in

y = 2

as though it were the

y

-axis.

y = 2

is a horizontal line through

(0, 2)

Omitting the centre of rotation or enlargement.

Always give the centre as coordinates.

Calling a scale factor

\frac{1}{2}

enlargement a "reduction" with no details.

"Enlargement, scale factor

\frac{1}{2}

, centre …"

Stretch yourself

A point $P$ is rotated $90^\circ$ anticlockwise about the origin, then reflected in the line $y = x$ . The final image is $(3, 5)$ . Find $P$ , and describe the single transformation that maps $P$ to its final image.

Hint — Work backwards: undo the reflection first, then undo the rotation.

Reveal solution

SolutionUndo the reflection in

y = x

(5, 3)

. Undo the

90^\circ

anticlockwise rotation by rotating

90^\circ

clockwise:

(x, y) \to (y, -x)

gives

(3, -5)

. So

P = (3, -5)

. Overall

(x, y) \to (-y, x) \to (x, -y)

: the single transformation is a reflection in the

x

-axis. Check:

(3, -5) \to (3, 5)

✓.

\blacksquare

Questions students ask

Is an enlargement with scale factor

\frac{1}{2}

really an enlargement?

Yes. Mathematically, any scale factor gives an enlargement, even if the shape gets smaller.

What does a negative scale factor do?

It enlarges through the centre, so the image is on the opposite side and appears rotated

180^\circ

Which transformations keep a shape congruent?

Translations, reflections and rotations. Enlargements (with scale factor not

\pm 1

) produce similar shapes instead.

Key takeaways

Translation: column vector $\begin{pmatrix} \text{right} \\ \text{up} \end{pmatrix}$ .
Reflection: equal perpendicular distances from the mirror line.
Rotation: angle, direction, centre.
Enlargement: scale factor and centre; negative factors go through the centre.
Describe fully, as a single transformation.

Exam-style questions

Practise it the way it is examined

3 original questions · 8 marks, written to match the style and mark allocation of the real papers. Work on paper, then open the mark scheme and tick the marks you earned — method marks count even if the final answer slips.

Question 1Grade 22 marks

Triangle $T$ has vertices $(1, 1)$ , $(4, 1)$ and $(1, 3)$ . Write down the vertices of the image of $T$ after a translation by $\begin{pmatrix} -2 \\ 3 \end{pmatrix}$ .[2]
Show mark scheme
- B1Two vertices correct
- B1 $(-1, 4)$ , $(2, 4)$ , $(-1, 6)$

Question 2Grade 53 marks

Shape $A$ has vertices $(2, 1)$ , $(4, 1)$ , $(4, 2)$ . Shape $B$ has vertices $(-1, 2)$ , $(-1, 4)$ , $(-2, 4)$ . Describe fully the single transformation that maps $A$ onto $B$ .[3]
Show mark scheme
- B1Rotation
- B1 $90^\circ$ anticlockwise
- B1Centre $(0, 0)$
Answer: Rotation, $90^\circ$ anticlockwise, centre $(0, 0)$

Examiner note

Check with the rule $(x, y) \to (-y, x)$ : $(2, 1) \to (-1, 2)$ ✓.

Question 3Grade 73 marks

Triangle $P$ has vertices $(2, 2)$ , $(6, 2)$ and $(2, 4)$ . It is enlarged by scale factor $-\frac{1}{2}$ with centre $(0, 0)$ to give triangle $Q$ . Write down the vertices of $Q$ , and the ratio of the area of $P$ to the area of $Q$ .[3]
Show mark scheme
- B1Vertices $(-1, -1)$ , $(-3, -1)$ , $(-1, -2)$
- M1Area scale factor $\left(\frac{1}{2}\right)^2 = \frac{1}{4}$
- A1 $4 : 1$

Unlimited practice

Skill drill

A new question every time, marked instantly, with a full worked solution. Aim for a streak of five, then move up a level.

0 / 0 first-time correct Streak 0Best at this level: 0

Generating a question…

Test yourself

Answer a few questions, then review anything you missed.

Prefer paper? Open the written self-check

Answer each question before opening its answer. This self-check does not record a quiz score.

Translate $(2, 5)$ by $\begin{pmatrix} 3 \\ -1 \end{pmatrix}$ .
- $(5, 4)$
- $(-1, 6)$
- $(5, 6)$
- $(6, -5)$
Show answer and explanation
Answer: $(5, 4)$
Add 3 to $x$ ; subtract 1 from $y$ .
Reflect $(3, 4)$ in the $x$ -axis.
- $(3, -4)$
- $(-3, 4)$
- $(4, 3)$
- $(-3, -4)$
Show answer and explanation
Answer: $(3, -4)$
The $y$ -coordinate changes sign.
Which transformation changes the size of a shape?
- Enlargement
- Rotation
- Reflection
- Translation
Show answer and explanation
Answer: Enlargement
The others keep shapes congruent.
Rotate $(2, 3)$ $90^\circ$ anticlockwise about the origin.
- $(-3, 2)$
- $(3, -2)$
- $(-2, -3)$
- $(3, 2)$
Show answer and explanation
Answer: $(-3, 2)$
$(x, y) \to (-y, x)$ .
Reflect $(1, 6)$ in $y = x$ .
- $(6, 1)$
- $(-1, 6)$
- $(1, -6)$
- $(-6, -1)$
Show answer and explanation
Answer: $(6, 1)$
Swap the coordinates.
A description of a rotation needs…
- angle, direction and centre
- a column vector
- a mirror line
- a scale factor and centre
Show answer and explanation
Answer: angle, direction and centre
All three are required.
Enlarge $(3, 1)$ by scale factor 2, centre $(1, 1)$ .
- $(5, 1)$
- $(6, 2)$
- $(4, 1)$
- $(7, 1)$
Show answer and explanation
Answer: $(5, 1)$
Vector $\binom{2}{0}$ becomes $\binom{4}{0}$ ; $(1 + 4, 1)$ .
An enlargement has scale factor $-1$ , centre $(0, 0)$ . It is the same as…
- a rotation of $180^\circ$ about $(0, 0)$
- a reflection in $y = x$
- a translation
- a reflection in the $x$ -axis
Show answer and explanation
Answer: a rotation of $180^\circ$ about $(0, 0)$
$(x, y) \to (-x, -y)$ .

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