Rotational Symmetry, Translation, Reflection and Rotation

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This page covers two related ideas. First, rotational symmetry: does a shape look exactly the same when you turn it part of the way round? Then the three transformations you need at Stage 6: translation (slide), reflection (flip) and rotation (turn). In every transformation the starting shape is called the object and the new shape is called the image. The image is always exactly the same size and shape as the object — we say the two shapes are congruent.

Key words

1. Rotational symmetry and its order

A shape has rotational symmetry if it fits exactly onto its own outline when you turn it around its centre before it has made a full turn. The order of rotational symmetry is the number of times it fits onto itself during one full turn (360°). The last position, after a full turn, is counted too — that is why the smallest possible order is 1.

How to find the order with tracing paper (the Checkpoint test allows tracing paper):

1. Trace the shape. Put a small arrow or dot in one corner of your tracing so you can see how far you have turned.
2. Put your pencil point on the centre of the shape and turn the tracing paper slowly.
3. Count every time the tracing fits exactly on top of the shape again.
4. Stop when the tracing is back where it started (one full turn). That last fit counts. Your total is the order.

Without tracing paper: look for a part of the shape that repeats. If the same part appears 4 times around the centre, all the same size, colour and direction, the order is 4. If it appears twice, opposite each other, the order is 2. Colour counts: if one part is shaded differently, the shape does not fit onto itself in that position.

2. Translation (sliding a shape)

In a translation, every point of the shape slides the same distance in the same direction. The shape does not turn and does not flip — the image looks exactly like the object, the same way up, just in a new place. We describe a translation with two moves: how many squares left or right, and how many squares up or down.

Translating with coordinates. Moving right adds to the x-coordinate, moving left subtracts from it. Moving up adds to the y-coordinate, moving down subtracts from it.

1. Choose one vertex. Count the squares across first (right or left), then up or down. Mark the new point.
2. Do the same for every other vertex — always with the same two moves.
3. Join the new points in the same order and label them A′, B′, C′ (say “A dash” or “A prime”).
4. Check: the image must be the same size, the same shape and the same way up as the object.

Describing a translation. Pick one pair of corresponding points and count from the object to the image: first across, then up or down. For example, from A(−5, 2) to A′(2, −3): x goes from −5 to 2, that is 7 right; y goes from 2 to −3, that is 5 down.

3. Reflection in a mirror line

In a reflection, the shape is flipped over a mirror line. Every point of the image is the same distance from the mirror line as the matching point of the object, but on the other side, and you measure that distance at right angles to the mirror line. A point that lies on the mirror line stays where it is. The image is reversed, like your reflection in a real mirror.

1. Take one vertex of the object. Count the squares from the vertex to the mirror line, moving at right angles to the line (straight across for a vertical line, straight up or down for a horizontal line).
2. Count the same number of squares on the other side of the line and mark the image point.
3. Repeat for every vertex. Vertices that are on the mirror line do not move.
4. Join the image points with a ruler. Check: if you folded the paper along the mirror line, the object would land exactly on the image.

Diagonal mirror lines. When the mirror line goes through the corners of the grid squares at 45°, you cannot count straight across. Instead, move from the vertex to the line diagonally, through the corners of the squares, at right angles to the mirror line. Then carry on the same number of diagonal steps on the other side. Another good method is to turn the page until the mirror line is vertical — then the reflection looks like an ordinary one.

4. Rotation by 90° around a vertex

In a rotation, the shape turns around a fixed point called the centre of rotation. At Stage 6 the turn is always 90° (a quarter turn), clockwise or anticlockwise, and the centre is one of the shape’s own vertices. The centre vertex does not move. Every side that starts at the centre turns through 90° and keeps its length.

Method 1 — tracing paper:

1. Trace the shape and draw a small arrow on the tracing pointing straight up from the centre vertex.
2. Hold the centre vertex down with your pencil point.
3. Turn the tracing a quarter turn in the direction asked, until your arrow points right (clockwise) or left (anticlockwise).
4. Press through the corners or note where they are, then draw the image on the grid with a ruler.

Method 2 — the “arms” method (no tracing paper). From the centre vertex, write each other vertex as moves across and up/down. Then turn each move a quarter turn:

Think of a clock: the hands move clockwise from 12 to 3, so up turns into right. For example, a vertex that is 3 right and 2 up from the centre becomes 3 down and 2 right after a clockwise quarter turn.

Worked examples

Worked example 1 — Rotational symmetry [1 mark] Tick (✓) the two shapes that have rotational symmetry of order 2: rhombus, kite, rectangle, isosceles trapezium.

1. Imagine a half turn of each shape. A rhombus and a rectangle both look the same upside down, so they fit after a half turn.
2. Check a quarter turn: a rhombus that is not a square does not fit after a quarter turn, nor does a rectangle that is not a square. So both have order exactly 2.
3. A kite and an isosceles trapezium only fit after a full turn: order 1.

Answer: rhombus ✓ and rectangle ✓ (both needed for the mark). Common mistake: counting lines of symmetry instead. A rectangle has 2 lines of symmetry and order 2 — but a parallelogram has 0 lines of symmetry and still has order 2.

Worked example 2 — Translation with coordinates [2 marks] Triangle PQR has vertices P(−3, 2), Q(−1, 2) and R(−3, 5). It is translated 4 squares right and 6 squares down. Write the coordinates of P′, Q′ and R′.

Answer: P′(1, −4), Q′(3, −4), R′(1, −1). Award 2 marks for all three correct, 1 mark for two correct. Check: PQ is horizontal and 2 units long; P′Q′ is also horizontal and 2 units long. ✓ Common mistake: adding the 4 to the y-coordinate. Right/left changes x (the first number); up/down changes y (the second number).

Worked example 3 — Describing a translation [1 mark] A shape is translated so that the point A(2, −1) moves to A′(−3, 2). Describe the translation.

1. Across: x goes from 2 to −3. Count on a number line: 2 → 0 is 2 steps, 0 → −3 is 3 more steps, so 5 steps to the left.
2. Up/down: y goes from −1 to 2. −1 → 0 → 2 is 3 steps up.
3. Check with any other corresponding pair — the moves are always the same in a translation.

Answer: 5 squares left and 3 squares up. Common mistake: counting from the image back to the object, which gives the opposite directions (5 right, 3 down).

Worked example 4 — Reflection in a horizontal mirror line [1 mark] The shape has vertices at (1, 4), (4, 4), (4, 6) and (2, 7) on a square grid. The mirror line is horizontal and passes through the two bottom vertices. Draw the reflection.

1. The vertices (1, 4) and (4, 4) are on the mirror line, so they stay where they are.
2. (4, 6) is 2 squares above the line → its image is 2 squares below the line: (4, 2).
3. (2, 7) is 3 squares above the line → its image is 3 squares below: (2, 1).
4. Join (1, 4), (4, 4), (4, 2), (2, 1) in order with a ruler.

Worked example 5 — Rotation about a vertex [2 marks] Triangle JKL has vertices J(1, 1), K(4, 1) and L(4, 3). It is rotated 90° anticlockwise about vertex J. Write the coordinates of K′ and L′.

1. J is the centre, so J does not move: J′ = (1, 1).
2. From J to K is 3 right. Anticlockwise, right turns into up → 3 up from J: K′ = (1, 4).
3. From J to L is 3 right and 2 up. Anticlockwise, right turns into up and up turns into left → 3 up and 2 left from J: L′ = (1 − 2, 1 + 3) = (−1, 4).
4. Check: JK and JK′ are both 3 units long and meet at a right angle. ✓

Answer: K′(1, 4), L′(−1, 4). 1 mark for each. Common mistake: turning clockwise instead. Clockwise would send K to (1, −2) — always check the direction word in the question.

Worked example 6 — Which transformation? [2 marks] Shape A has vertices (1, 1), (3, 1), (1, 4). Shape B has vertices (5, 3), (7, 3), (5, 6). Shape C has vertices (−1, 1), (−3, 1), (−1, 4). (a) Describe fully the transformation that maps shape A onto shape B. (b) Describe fully the transformation that maps shape A onto shape C.

1. (a) Compare corresponding points: (1, 1) → (5, 3), (3, 1) → (7, 3), (1, 4) → (5, 6). Each x goes up by 4 and each y goes up by 2. The shape faces the same way, so it is a translation of 4 squares right and 2 squares up.
2. (b) Each y-coordinate stays the same, but each x-coordinate changes sign: 1 → −1, 3 → −3. The shape is flipped across the y-axis, so it is a reflection in the y-axis (the vertical line through 0).

Exam technique: “Describe fully” means name the transformation and give the details: for a translation the moves; for a reflection the mirror line; for a rotation the angle, the direction and the centre.

Practice

All pages in this course

Start here

• Checkpoint 6 Maths: The Test and How to Prepare

Number: integers and calculation

• Counting On and Back and Negative Numbers
• Sequences, Letters and Position-to-Term Rules
• Order of Operations, Brackets and Laws of Arithmetic
• Multiplying Whole Numbers (up to 10 000 × 2-digit)
• Dividing Whole Numbers (up to 1000 ÷ 2-digit)
• Factors, Multiples and Tests of Divisibility
• Square Numbers and Cube Numbers

Number: place value, decimals, fractions, percentages and ratio

• Place Value and Decimals to Thousandths
• Rounding Decimals
• Fractions: Division, Operators and Simplest Form
• Adding and Subtracting Fractions
• Multiplying and Dividing Fractions by Whole Numbers
• Fractions, Decimals and Percentages: Equivalence and Ordering
• Adding, Subtracting, Multiplying and Dividing Decimals
• Ratio and Direct Proportion

Geometry and measure

• Time: Decimals and Mixed Units
• Quadrilaterals: Properties and Classification
• Circles: Parts and Constructing Circles
• Angles: Measuring, Drawing and Angles in a Triangle
• Area of Rectangles and Right-angled Triangles
• 3D Shapes, Compound Shapes and Nets
• Capacity, Volume and Surface Area
• Coordinates in Four Quadrants
• Rotational Symmetry, Translation, Reflection and Rotation

Statistics and probability

• Collecting and Representing Data
• Mode, Median, Mean and Range, and Interpreting Data
• Probability

Practice tests

• Checkpoint 6 Maths — Practice Test 1
• Checkpoint 6 Maths — Practice Test 2
• Checkpoint 6 Maths — Practice Test 3