Translation, Reflection, Rotation and Enlargement

British & IGCSE › Checkpoint 8 › Checkpoint 8 Maths — Complete Course

A transformation moves or changes a shape. The starting shape is the object; the new shape is the image. If the object is A, we often call the image A′ (say “A dash”). At Stage 8 you must be able to do four transformations on a coordinate grid and describe each one fully — that means giving every piece of information somebody else would need to repeat it exactly.

Key words

1. Translation with column vectors

In a translation every point of the shape moves the same distance in the same direction. The shape does not turn or flip, so the image is congruent to the object and points the same way. At Stage 8 we describe a translation with a column vector: two numbers written one above the other in tall brackets.

On this web page we type a column vector on one line as (3 over −2) — top number 3, bottom number −2. In your test paper, always write it as a column, top number above bottom number, inside tall brackets. Never write a vector like a coordinate (3, −2): coordinates are a position, a vector is a movement.

1. Read the vector: top number = move across (positive → right, negative → left); bottom number = move up or down (positive → up, negative → down).
2. Using coordinates: add the top number to every x-coordinate and add the bottom number to every y-coordinate.
3. Plot the new vertices and join them in the same order. Label the image (e.g. A′).
4. Check: the image must be the same size and the same way round as the object.

Finding the vector of a translation. Choose one vertex of the object and the matching vertex of the image. Vector = (image x − object x) over (image y − object y). Always work from the object to the image. Going back from A′ to A uses the opposite vector (−6 over 4).

2. Reflection in a mirror line

In a reflection, each point of the image is the same perpendicular distance from the mirror line as the matching point of the object, but on the other side. Points on the mirror line do not move. The image is congruent to the object but flipped (its vertices go round the other way).

Equations of mirror lines. Every point on a vertical line has the same x-coordinate, so the line is x = a number. Every point on a horizontal line has the same y-coordinate, so the line is y = a number. The x-axis is the line y = 0 and the y-axis is the line x = 0. The two diagonal lines through the origin are y = x (points like (2, 2), (−3, −3)) and y = −x (points like (2, −2), (−3, 3)).

You do not need to learn the last two rows as formulas: just count. For the line x = 1, the point (3, 1) is 2 squares to the right of the line, so its image is 2 squares to the left of the line, at x = −1.

Reflecting in a diagonal line on a grid. Move from each vertex to the mirror line diagonally across the squares (at right angles to the line), then the same number of diagonal steps on the other side. For y = x you can also just swap the coordinates: (2, −1) → (−1, 2), (4, −1) → (−1, 4), (4, 1) → (1, 4).

Identifying a reflection. If a shape and its image are congruent but the image is flipped, it is a reflection. To find the mirror line, take a vertex and its image: the mirror line goes through the midpoint of the two points and is at right angles to the line joining them. Example: (2, 5) and its image (2, −1). The midpoint is (2, 2), and the points are one above the other, so the mirror line is horizontal: y = 2.

3. Rotation: centre, angle and direction

In a rotation, the shape turns around a fixed point, the centre of rotation. You need three facts to rotate a shape or to describe a rotation — if one is missing, the rotation is not fully described:

• the centre of rotation, as coordinates, e.g. (−1, 0) — it can be inside, on or outside the shape;
• the angle: usually 90° (a quarter turn), 180° (a half turn) or 270° (three quarters of a turn);
• the direction: clockwise (the way clock hands turn) or anticlockwise. A 180° turn is the same in both directions, so the direction is not needed for 180°.

Method 1 — tracing paper (allowed in the test). Trace the object and mark the centre on the tracing. Put your pencil point on the centre, turn the tracing through the angle in the given direction (a quarter turn for 90°), then press through or copy the vertices on to the grid underneath.

Method 2 — the “arm” method. From the centre, describe each vertex as a move across and a move up or down. Turn that move through the angle: in a 90° clockwise turn, up becomes right, right becomes down, down becomes left and left becomes up. Then step out from the centre with the new move.

Identifying a rotation. If the image is congruent, not flipped, but pointing a different way, it is a rotation. Look at how one side has turned to find the angle and direction (a horizontal side that becomes vertical has turned 90°). Find the centre by trying points with tracing paper: the correct centre is the point that, when you hold it still and turn the tracing, takes the object exactly on to the image. The centre is the same distance from a vertex and from that vertex’s image.

4. Enlargement: centre and scale factor

An enlargement changes the size of a shape. Every length is multiplied by the scale factor, and all angles stay the same, so the image is similar to the object (same shape, different size) — not congruent. At Stage 8 the scale factor is a positive whole number (2, 3, 4…). The centre of enlargement is the fixed point that all distances are measured from; it can be outside the shape or on it.

1. For one vertex, count how far it is from the centre: across and up/down (e.g. 2 right, 1 up).
2. Multiply both distances by the scale factor (scale factor 2: 4 right, 2 up).
3. Count that far from the centre (not from the vertex!) and mark the image vertex.
4. Repeat for every vertex, join them up, and check: each side of the image should be (scale factor) × the matching side of the object.
5. Extra check: the centre, a vertex and its image are always on one straight line (a ray).

Centre on the shape. If the centre is a vertex of the shape, that vertex stays where it is and the shape grows away from it. For example, enlarging the square with vertices (1, 1), (2, 1), (2, 2), (1, 2) by scale factor 3 with centre (1, 1) gives the square (1, 1), (4, 1), (4, 4), (1, 4).

Identifying an enlargement. (1) Scale factor = length of a side on the image ÷ length of the matching side on the object. (2) Centre: draw straight lines through each image vertex and its matching object vertex, and extend them. They all meet at one point: the centre of enlargement.

Worked examples

Worked example 1 — Translation [2 marks] Triangle P has vertices (−4, 1), (−2, 1) and (−4, 4). It is translated by the vector (5 over −3). Write the coordinates of the vertices of the image.

1. Top number 5: add 5 to every x-coordinate. Bottom number −3: subtract 3 from every y-coordinate.
2. (−4, 1) → (−4 + 5, 1 − 3) = (1, −2)
3. (−2, 1) → (3, −2)
4. (−4, 4) → (1, 1)

Answer: (1, −2), (3, −2) and (1, 1). Mark scheme style: 2 marks for all three correct; 1 mark for two correct. Common mistake: moving 3 up because the 3 looks positive. The bottom number is −3, so the move is down.

Worked example 2 — Describing a translation [2 marks] Shape A is translated to shape B. The vertex (1, 2) of A moves to the vertex (−3, 5) of B. Describe fully the single transformation that maps A on to B.

1. Across: −3 − 1 = −4 (4 left).
2. Up/down: 5 − 2 = 3 (3 up).
3. A full description names the transformation and gives the vector.

Answer: a translation by the vector (−4 over 3) (1 mark for “translation”, 1 mark for the vector). Common mistake: writing “move 4 left and 3 up” without the word translation, or writing the vector as a coordinate (−4, 3).

Worked example 3 — Reflection in y = x and y = −x [2 marks] The point M is (5, −2). Write the coordinates of the image of M after (a) a reflection in the line y = x, (b) a reflection in the line y = −x.

1. (a) Reflection in y = x swaps the coordinates: (5, −2) → (−2, 5).
2. (b) Reflection in y = −x swaps the coordinates and changes both signs: (5, −2) → (2, −5).
3. Check (b): is M above or below the line y = −x? At x = 5 the line is at y = −5, and M is at y = −2, so M is above the line; its image (2, −5) is below it (at x = 2 the line is at y = −2). ✓

Answer: (a) (−2, 5) (b) (2, −5) Common mistake: in (b), swapping the numbers but forgetting to change the signs, giving (−2, 5) again.

Worked example 4 — Identifying a mirror line [1 mark] The point (−1, 3) is reflected to the point (5, 3). Write the equation of the mirror line.

1. The y-coordinates are the same, so the points are side by side and the mirror line is vertical: x = something.
2. The mirror line is halfway between them: x = (−1 + 5) ÷ 2 = 2.

Answer: x = 2 Common mistake: writing y = 2. A vertical line has an equation x = …, because every point on it has the same x-coordinate.

Worked example 5 — Rotation [2 marks] Triangle R has vertices (1, 1), (3, 1) and (1, 2). Rotate R 90° clockwise about the centre (−1, 0). Write the coordinates of the image. (See the diagram in section 3.)

Answer: (0, −2), (0, −4) and (1, −2). Check: the side from (1, 1) to (3, 1) was horizontal, length 2; the image side from (0, −2) to (0, −4) is vertical, length 2 ✓. Common mistake: rotating about the origin or about a vertex of the shape instead of the centre you were given.

Worked example 6 — Enlargement [2 marks] Triangle E has vertices (2, 2), (4, 2) and (2, 3). Enlarge E by scale factor 2 with centre (0, 1). (See the diagram in section 4.)

Answer: (4, 3), (8, 3) and (4, 5). Check: base 2 → 4 and height 1 → 2, both doubled ✓. Common mistake: making the shape twice as big but drawing it in the wrong place (for example, adding 2 to the coordinates). The position depends on the centre.

Worked example 7 — Identifying an enlargement [3 marks] In the diagram in section 4, shape A has vertices (2, 2), (3, 2), (3, 4) and shape B has vertices (4, 4), (7, 4), (7, 10). Describe fully the single transformation that maps A on to B.

1. B is bigger and the same shape, so it is an enlargement.
2. Scale factor: base of A = 3 − 2 = 1; base of B = 7 − 4 = 3. 3 ÷ 1 = 3. (Check the heights: 2 and 6 ✓.)
3. Centre: draw lines through (4, 4) and (2, 2), and through (7, 4) and (3, 2), and through (7, 10) and (3, 4). They meet at (1, 1).
4. Check with the vector method: (2, 2) is 1 right and 1 up from (1, 1); × 3 gives 3 right and 3 up → (4, 4) ✓.

Answer: an enlargement, scale factor 3, centre (1, 1) — 1 mark for each part. Common mistake: giving only “enlargement scale factor 3”. Without the centre the description is not full and loses a mark.

Practice

All pages in this course

Start here

• Stage 8 Maths: The Tests and How to Prepare

Number

• Order of Operations and Calculating with Integers
• Factors, Multiples, Primes, HCF and LCM
• Types of Number and Index Laws
• Squares, Cubes and Roots (Positive and Negative)
• Multiplying and Dividing by 0.1 and 0.01, and Significant Figures
• Fractions: Recurring Decimals, Mixed Numbers and Order of Operations
• Percentage Increase, Decrease and Absolute Change
• Ordering and Calculating with Decimals
• Ratio and Proportion

Algebra

• Algebraic Expressions: Simplifying, Expanding and Factorising
• Writing Expressions and Formulae, and Changing the Subject
• Equations and Inequalities
• Sequences and the nth Term
• Functions and Straight-line Graphs (y = mx + c)
• Reading and Interpreting Real-life Graphs

Geometry and measure

• Quadrilaterals and Regular Polygons
• Circumference of a Circle, and Miles and Kilometres
• Area of Parallelograms and Trapezia
• 3D Shapes: Euler's Formula and Plans and Elevations
• Volume of Prisms and Surface Area
• Angles in Triangles and on Parallel Lines
• Constructions with Ruler and Compasses
• Bearings and Midpoints
• Translation, Reflection, Rotation and Enlargement

Statistics and probability

• Data Collection and Sampling
• Representing Data
• Averages, Range and Comparing Distributions
• Probability: Complementary and Combined Events

Practice tests

• Stage 8 Maths — Practice Test 1
• Stage 8 Maths — Practice Test 2
• Stage 8 Maths — Practice Test 3