📖 Theory notes

Every note is built the same way: I can…, a quick recall, 🔑 key ideas, ✨ tips, ⚠️ common mistakes, ✏️ worked examples and 🧠 Remember in 3 lines.

🎯 Learn, then practise from two books

Each LO starts with 📘 a learning sheet (worked examples one step at a time), then 📗 Collins practice (Book 1 exercises) and, where Book 2 covers the topic, 📙 Pearson practice (Book 2 exercises, including starred questions).

🧪 LO tests · 30 minutes each

When the practice feels easy, take the timed test for that LO. Tests run in Quiz mode, submit themselves at 00:00 and can be taken once. Your report shows each LO as a pie chart.

About this chapter

A transformation moves a shape or changes its size. There are four you need: translation (slide), reflection (flip), rotation (turn) and enlargement (resize). The first three keep the shape congruent; an enlargement makes it similar. In the exam you must draw transformations on a grid and describe them fully.

LO	📗 Collins Book 1	📙 Pearson Book 2
LO 1	30.1 Translations	—
LO 2	30.2 Reflections	—
LO 3	30.3 Further reflections	—
LO 4	30.4–30.5 Rotations	—
LO 5	30.6 Enlargements	—

Book 2 (Units 6–10) does not cover transformations of shapes, so every LO uses Collins exercises only.

To describe…	you must give
a translation	the column vector
a reflection	the equation of the mirror line
a rotation	the centre, the angle and the direction
an enlargement	the scale factor and the centre

Learning objectives

LO 1 · 30.1 TranslationsI can describe a translation with a column vector and translate points and shapes by a given vector.

LO 2 · 30.2 Reflections in horizontal and vertical linesI can reflect points and shapes in lines like x = a, y = b and the axes, and find the equation of a mirror line.

LO 3 · 30.3 Reflections in y = x and y = −xI can reflect points and shapes in the lines y = x and y = −x and use the rules (a, b) → (b, a) and (a, b) → (−b, −a).

LO 4 · 30.4–30.5 RotationsI can rotate a shape about a centre through 90° or 180°, and describe a rotation fully by its centre, angle and direction.

LO 5 · 30.6 EnlargementsI can enlarge a shape by a whole-number or fractional scale factor about a centre, and find the scale factor and centre of an enlargement.

Check what you already know

You need	Quick check	Answer
Plotting coordinates	Which point is 3 right and 2 down from the origin?	(3, −2)
Equations of lines	The line through (4, 0) and (4, 5)	x = 4
Midpoints	Midpoint of (1, 6) and (5, −2)	(3, 2)
Angles of turn	A quarter turn is…	90°

30.1 Translations

LO 1I can… describe a translation with a column vector and translate points and shapes by a given vector.

🔁 Quick recall before you start
On a grid, what is the movement from (2, 1) to (5, −1)? (3 right and 2 down.)

📗 Collins: 30.1 Translations, Example 1, Exercise 30A
📙 Pearson Book 2: no matching unit — Collins practice only

🔑 Key idea · Column vectors
A translation slides every point the same distance in the same direction. We write it as a column vector: 3−2 means 3 right and 2 down.
Top number: across (right +, left −). Bottom number: up (+) or down (−).

✏️ Worked example · Collins Example 1

A to B: 21 B to C: 20 C to D: −32 D to A: −1−3
Follow one corner of the triangle from the object to the image.

✨ Tip · Coordinates
To translate a point, add the vector: (x, y) + ab = (x + a, y + b).

⚠️ Common mistake · Top and bottom
Students often swap the numbers. The top number is always the horizontal move.

✨ Tip · Going back
The reverse of ab is −a−b.

🧠 Remember in 3 lines

Translation = slide; the shape does not turn or flip.
Column vector: top = across, bottom = up/down.
Add the vector to every vertex.

Memory hook: Along the corridor, then up the stairs — across first, up second.

30.2 Reflections in horizontal and vertical lines

LO 2I can… reflect points and shapes in lines like x = a, y = b and the axes, and find the equation of a mirror line.

🔁 Quick recall before you start
Which line is x = 0? (The y-axis.) Which line is y = 0? (The x-axis.)

📗 Collins: 30.2 Reflections, Example 2, Exercise 30B
📙 Pearson Book 2: no matching unit — Collins practice only

🔑 Key idea · Mirror lines
A reflection flips a shape in a mirror line. Each point and its image are the same distance from the mirror, on opposite sides, and the line joining them is perpendicular to the mirror.

Mirror line	Rule
x-axis (y = 0)	(a, b) → (a, −b)
y-axis (x = 0)	(a, b) → (−a, b)
x = k	(a, b) → (2k − a, b)
y = k	(a, b) → (a, 2k − b)

✏️ Worked example · Reflect (1, 4)

In x = 2: 1 is 1 left of the mirror, so the image is 1 right: (3, 4).
In y = −1: 4 is 5 above, so the image is 5 below: (1, −6).

✏️ Worked example · Find the mirror line

(2, 3) maps to (6, 3). The mirror is halfway: x = 4.

⚠️ Common mistake · x = 2 is vertical
The line x = 2 goes up and down (all points have x = 2). Students often draw it across.

✨ Tip · Describe fully
To describe a reflection you only need the equation of the mirror line, e.g. “reflection in the line y = −1”.

🧠 Remember in 3 lines

Same distance, opposite side, at 90° to the mirror.
x = k is vertical; y = k is horizontal.
Mirror line = halfway between point and image.

Memory hook: Your mirror twin stands as far behind the glass as you stand in front.

30.3 Reflections in y = x and y = −x

LO 3I can… reflect points and shapes in the lines y = x and y = −x and use the rules (a, b) → (b, a) and (a, b) → (−b, −a).

🔁 Quick recall before you start
Plot y = x: does it go through (2, 2)? (Yes.) Does y = −x go through (3, −3)? (Yes.)

📗 Collins: 30.3 Further reflections, Example 3, Exercise 30C
📙 Pearson Book 2: no matching unit — Collins practice only

🔑 Key idea · Sloping mirrors
Reflection in y = x swaps the coordinates: (a, b) → (b, a).
Reflection in y = −x swaps and changes both signs: (a, b) → (−b, −a).

✏️ Worked example · Collins Example 3 (adapted)

t has vertices (4, 1), (7, 1), (7, 3).
Reflect in y = x: swap each pair → (1, 4), (1, 7), (3, 7).

✨ Tip · Turn the page
Turn the page so the mirror line looks vertical or horizontal, or count diagonal steps across the line.

⚠️ Common mistake · Points on the mirror
A point on the mirror line does not move: (5, 5) stays at (5, 5) under y = x.

✏️ Worked example · Chain of reflections

(a, b) → y = x → (b, a) → y = −x → (−a, −b).
Two reflections in perpendicular lines = a 180° rotation about where they cross.

🧠 Remember in 3 lines

y = x: swap.
y = −x: swap and change both signs.
Points on the mirror stay still.

Memory hook: y = x is a swap shop.

30.4–30.5 Rotations

LO 4I can… rotate a shape about a centre through 90° or 180°, and describe a rotation fully by its centre, angle and direction.

🔁 Quick recall before you start
How many degrees is a quarter turn? (90°.) A half turn? (180°.)

📗 Collins: 30.4–30.5 Rotations, Example 4, Exercises 30D and 30E
📙 Pearson Book 2: no matching unit — Collins practice only

🔑 Key idea · Centre, angle, direction
A rotation turns a shape about a fixed point, the centre of rotation. Give the angle and the direction: anticlockwise is positive, clockwise is negative (−90° = 90° clockwise). 180° is the same either way.

About the origin	Rule
90° clockwise (−90°)	(x, y) → (y, −x)
180°	(x, y) → (−x, −y)
90° anticlockwise (+90°)	(x, y) → (−y, x)

✏️ Worked example · About another centre

Rotate (4, 3) by 90° anticlockwise about (1, 1).
Centre → point: 32. Turn it: −23.
Image = (1 − 2, 1 + 3) = (−1, 4).

✏️ Worked example · Collins Exercise 30E Q6

A: (1, 4), (3, 4), (3, 1) → B: (0, −2), (−3, −2), (−3, −4).
Rotation 90° anticlockwise about (3, −2).

✨ Tip · Finding the centre
Use tracing paper and try likely points. For a 180° turn the centre is the midpoint of any point and its image.

⚠️ Common mistake · Incomplete descriptions
“Rotation of 90°” loses marks. Write “rotation 90° clockwise about (3, −2)”.

🧠 Remember in 3 lines

Centre + angle + direction.
Anticlockwise is +, clockwise is −.
Half turn: centre = midpoint of point and image.

Memory hook: CAD — Centre, Angle, Direction.

30.6 Enlargements

LO 5I can… enlarge a shape by a whole-number or fractional scale factor about a centre, and find the scale factor and centre of an enlargement.

🔁 Quick recall before you start
Multiply the vector 23 by 3. (69.)

📗 Collins: 30.6 Enlargements, Examples 5–8, Exercise 30F
📙 Pearson Book 2: no matching unit — Collins practice only

🔑 Key idea · Scale factor and centre
An enlargement multiplies every length by the scale factor and every distance from the centre of enlargement by the same number. Angles do not change, so the image is similar to the object.

✏️ Worked example · Collins Example 7

Enlarge A(2, 1), B(3, 1), C(3, 5) by scale factor 3 from (1, 2).
Centre → C: 23 × 3 = 69, so C′ = (7, 11).
Likewise A′(4, −1) and B′(7, −1).

🔑 Key idea · Fractional scale factors
A scale factor between 0 and 1 makes the image smaller and closer to the centre — it is still called an enlargement.

✨ Tip · Describing an enlargement
Scale factor = image length ÷ object length. Draw rays through matching vertices; they meet at the centre.

Scale factor k	Lengths	Perimeter	Area
2	× 2	× 2	× 4
3	× 3	× 3	× 9
⅓	÷ 3	÷ 3	÷ 9

⚠️ Common mistake · Adding instead of multiplying
Scale factor 3 does not mean “add 3 squares”. Multiply the distances from the centre by 3.

🧠 Remember in 3 lines

Multiply every distance from the centre by k.
Angles stay; lengths × k; areas × k².
Rays through matching points meet at the centre.

Memory hook: Shine a torch from the centre — the shadow grows along the rays.