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 $(a b)$ 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.

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 $(6 - 7)$ . Give the coordinates of its image.

Your answer

Type coordinates like (3, -1).

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).

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: $9 0^{\circ}$ clockwise sends $(x, y) \to (y, - x)$ ; $9 0^{\circ}$ anticlockwise sends $(x, y) \to (- y, x)$ ; $18 0^{\circ}$ sends $(x, y) \to (- x, - y)$ .

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

Answer

(1, - 4)

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

Rotate the point $(- 3, 2)$ through $18 0^{\circ}$ about the origin. Give the image.

Your answer

Type coordinates like (3, -1).

Tip — A $18 0^{\circ}$ 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.

1From the centre to the point:

(32)

2Multiply by 3:

(96)

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).

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, $9 0^{\circ}$ 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.

1Reflect in

x

-axis:

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

2Then in

y

-axis:

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

3That is the effect of a

18 0^{\circ}

rotation about the origin.

AnswerRotation,

18 0^{\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

Translation: (a b)

The column vector.

Reflection: mirror line

Such as

y = - x

x = 2

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

(3 - 2)

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 $9 0^{\circ}$ 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.

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

18 0^{\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 $(right up)$ .
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 $(- 2 3)$ .[2]

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]

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]

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Transformations

Translations and reflections

Rotations

Enlargements

Describing and combining transformations

Practise it the way it is examined

Skill drill

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Transformations

Translations and reflections

Rotations

Enlargements

Describing and combining transformations

Practise it the way it is examined

Skill drill

Practise this topic

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