1MA1 · G7–G8, G24Geometry & MeasuresFoundation & HigherGrades 2–7

Transformations

A transformation moves or resizes a shape according to a rule. There are four at GCSE — translation, reflection, rotation and enlargement — and each needs specific information to describe it completely. "Describe fully the single transformation" questions are worth two or three marks, and most lost marks are for missing a detail such as the centre of rotation.

50 min Transformations & Vectors

The big picture

Translations, reflections and rotations keep a shape exactly the same size and shape: the image is congruent to the original. Enlargements change size but keep the shape: the image is similar. That distinction links this lesson directly to congruence and similarity.

Each transformation has a checklist. Translation: a column vector. Reflection: the mirror line. Rotation: angle, direction and centre. Enlargement: scale factor and centre. If your description has all the items on the checklist, it earns full marks.

What you'll learn

Translate shapes using column vectors
Reflect shapes in lines such as $x = a$ , $y = b$ and $y = \pm x$
Rotate shapes about a centre, and find centres of rotation
Enlarge shapes with positive, fractional and negative scale factors, and describe transformations fully

Translations

A translation slides every point the same distance in the same direction, described by a column vector $\begin{pmatrix} x \\ y \end{pmatrix}$ : $x$ units right (negative for left) and $y$ units up (negative for down).

To describe a translation, pick one vertex and its image, and count across and up.

\begin{pmatrix} 3 \\ -2 \end{pmatrix}: \text{ 3 right, 2 down}

Column vector for a translation.

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

Translate the point $(3, -2)$ by $\begin{pmatrix} -5 \\ 4 \end{pmatrix}$ .

Your answer

Type coordinates like (2, -3).

Show worked solution

Solution

1 $3 - 5 = -2$ and $-2 + 4 = 2$ .

Answer: $(-2, 2)$

Reflections and rotations

A reflection flips a shape in a mirror line. Each point and its image are the same perpendicular distance from the line. Common lines: $x = a$ (vertical), $y = b$ (horizontal), $y = x$ and $y = -x$ (diagonal).

A rotation turns a shape about a fixed $\textbf{centre}$ through an $\textbf{angle}$ in a $\textbf{direction}$ (clockwise or anticlockwise). Tracing paper helps: pin it at the centre and turn.

To find a centre of rotation, try points with tracing paper, or construct perpendicular bisectors of lines joining two points to their images — the centre is where they cross.

Example 1Triangle

A

has vertices

(1, 1)

(3, 1)

and

(1, 2)

. Find the vertices after reflection in

y = x

1Reflecting in

y = x

swaps the coordinates:

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

Answer

(1, 1)

(1, 3)

and

(2, 1)

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

Rotate the point $(2, 5)$ through $90^\circ$ clockwise about the origin.

Your answer

Type coordinates like (2, -3).

Show worked solution

Solution

1 $90^\circ$ clockwise: $(x, y) \to (y, -x)$ .
2 $(5, -2)$ .

Answer: $(5, -2)$

Tip — A rotation of $180^{\circ}$ needs no direction, but $90^{\circ}$ does. "Rotation $90^{\circ}$ about $(0, 0)$ " without clockwise or anticlockwise loses a mark.

Enlargements

An enlargement multiplies distances from a $\textbf{centre}$ by the $\textbf{scale factor}$ $k$ . Draw rays from the centre through each vertex; the image vertex is $k$ times as far along.

A scale factor between 0 and 1 makes the shape smaller but is still called an enlargement.

A negative scale factor puts the image on the opposite side of the centre, upside down: with $k = -2$ , each image point is twice as far from the centre in the opposite direction.

Example 1Enlarge the point

P(4, 3)

by scale factor

-\tfrac12

with centre

(2, 1)

1Vector from centre to

P

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

2Multiply by

-\tfrac12

\begin{pmatrix} -1 \\ -1 \end{pmatrix}

3Add to the centre.

Answer

P^{\prime} = (1, 0)

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

Enlarge the point $(5, 3)$ by scale factor 2 with centre $(1, 1)$ .

Your answer

Type coordinates like (2, -3).

Show worked solution

Solution

1Vector from centre: $\begin{pmatrix} 4 \\ 2 \end{pmatrix}$ , doubled: $\begin{pmatrix} 8 \\ 4 \end{pmatrix}$ .
2Image: $(1 + 8, 1 + 4) = (9, 5)$ .

Answer: $(9, 5)$

Why this works

Working with the vector from the centre turns enlargement into arithmetic: multiply the vector by $k$ and add it to the centre. It works for any scale factor, including negatives.

Describing and combining transformations

Checklist for "describe fully": translation — vector; reflection — equation of mirror line; rotation — angle, direction, centre; enlargement — scale factor, centre.

A single transformation only: do not describe two transformations when one is asked for.

Combined transformations can often be replaced by one. Two reflections in parallel lines give a translation; two reflections in perpendicular lines through a point give a $180^{\circ}$ rotation about that point.

Example 1Shape

B

is the image of shape

A

under a reflection in the

x

-axis followed by a reflection in the

y

-axis. Describe the single transformation that maps

A

B

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

2That is the effect of a half turn about the origin.

AnswerRotation of

180^{\circ}

about

(0, 0)

Think like an examiner

Use the full checklist of details for each transformation.
Use the word "enlargement" even for scale factors less than 1.
Describe one single transformation when asked.

Transformations

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

Reflection in y = x.

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

Rotation 180° about the origin.

P^{\prime} = C + k(P - C)

Enlargement, centre C, factor k.

Watch out for these

Describing a rotation without its centre.

Give angle, direction and centre.

Writing a translation as coordinates

(3, -2)

Use a column vector.

Calling a scale factor

\tfrac12

a "reduction".

It is still an enlargement with scale factor

\tfrac12

Placing a negative enlargement on the same side of the centre.

Negative factors put the image on the opposite side.

Stretch yourself

Triangle $T$ has vertices $(2, 1)$ , $(4, 1)$ and $(2, 4)$ . It is rotated $90^{\circ}$ anticlockwise about the origin, then reflected in the line $y = x$ . Find the image vertices and describe the single transformation equivalent to the combination.

Hint — A $90^{\circ}$ anticlockwise rotation about the origin maps $(x, y) \to (-y, x)$ . Reflection in $y = x$ swaps coordinates.

Reveal solution

SolutionRotation:

(2, 1) \to (-1, 2)

(4, 1) \to (-1, 4)

(2, 4) \to (-4, 2)

. Reflection in

y = x

(-1, 2) \to (2, -1)

(-1, 4) \to (4, -1)

(-4, 2) \to (2, -4)

. Overall

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

, so every point keeps its

x

-coordinate and its

y

-coordinate changes sign: the single transformation is a reflection in the

x

-axis (

y = 0

\blacksquare

Questions students ask

Which transformations give congruent shapes?

Translations, reflections and rotations. Enlargements give similar shapes (congruent only if the scale factor is 1 or −1).

How do I find the centre of an enlargement?

Draw lines through corresponding vertices of the object and image; they meet at the centre.

Is a rotation of 90° clockwise the same as 270° anticlockwise?

Yes, they have the same effect.

Key takeaways

Translation: column vector.
Reflection: equation of the mirror line.
Rotation: angle, direction and centre.
Enlargement: scale factor and centre; negatives put the image on the far side.
Combined transformations can often be replaced by a single one.

Exam-style questions

Practise it the way it is examined

3 original questions · 7 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 32 marks

A triangle has vertices $(1, 2)$ , $(3, 2)$ and $(3, 5)$ . Write down the vertices of its reflection in the $y$ -axis.[2]
Show mark scheme
- B1Two correct
- B1 $(-1, 2)$ , $(-3, 2)$ , $(-3, 5)$

Question 2Grade 53 marks

Triangle $A$ has vertices $(1, 1)$ , $(3, 1)$ , $(1, 4)$ . Triangle $B$ has vertices $(2, 2)$ , $(6, 2)$ , $(2, 8)$ . Describe fully the single transformation that maps $A$ onto $B$ .[3]
Show mark scheme
- B1Enlargement
- B1Scale factor 2
- B1Centre $(0, 0)$

Question 3Grade 72 marks

A shape is rotated $90^\circ$ anticlockwise about the origin, then reflected in the line $y = x$ . Describe the single transformation that has the same effect.[2]
Show mark scheme
- M1Tracks a point, e.g. $(x, y) \to (-y, x) \to (x, -y)$
- A1Reflection in the $x$ -axis

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.

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Test yourself

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