Symmetry and Transformations — KS3 Maths Explained

Symmetry describes when a shape looks the same after a reflection or rotation. Transformations move or resize a shape: reflections flip it, rotations turn it, translations slide it, and enlargements change its size. All four transformations are tested at KS3 and GCSE.

Line symmetry

A shape has line symmetry if a mirror line can be drawn through it so that one half is an exact reflection of the other.

The number of lines of symmetry depends on the shape:

Shape | Lines of symmetry
Equilateral triangle | 3
Square | 4
Rectangle | 2
Regular pentagon | 5
Regular hexagon | 6
Parallelogram (non-rectangle) | 0
Isosceles triangle | 1
Scalene triangle | 0

To test for a line of symmetry, fold the shape (or imagine folding it). If both halves align exactly, the fold line is a line of symmetry.

Rotational symmetry

A shape has rotational symmetry if it looks identical after being rotated by less than a full turn (360°) about its centre.

The order of rotational symmetry is the number of times the shape looks the same during one full rotation.

Shape | Order of rotational symmetry
Equilateral triangle | 3
Square | 4
Rectangle | 2
Regular hexagon | 6
Parallelogram | 2
Scalene triangle | 1 (no rotational symmetry)

A shape with order 1 has no rotational symmetry — it only looks the same after a full 360° turn.

The four transformations

At KS3 you need to know four transformations: reflection, rotation, translation, and enlargement.

1. Reflection

A reflection flips a shape across a mirror line. Every point on the image is the same perpendicular distance from the mirror line as the corresponding point on the original, but on the opposite side.

What to state when describing a reflection: the mirror line (e.g. "the line y = 2" or "the y-axis").

Worked example 1 — reflect a triangle in the y-axis

Triangle with vertices at A(2, 1), B(5, 1), C(3, 4).

Reflecting in the y-axis: change the sign of every x-coordinate.

Image vertices: A'(−2, 1), B'(−5, 1), C'(−3, 4).

The shape is the same size and shape — only its position relative to the y-axis has flipped.

2. Rotation

A rotation turns a shape by a given angle about a centre of rotation, in a given direction (clockwise or anticlockwise).

What to state when describing a rotation: angle, direction, and centre of rotation.

Worked example 2 — rotate 90° anticlockwise about the origin

Point (3, 1) rotated 90° anticlockwise about the origin becomes (−1, 3).

Rule for 90° anticlockwise: (x, y) → (−y, x)

Rule for 90° clockwise: (x, y) → (y, −x)

Rule for 180°: (x, y) → (−x, −y)

3. Translation

A translation slides a shape by a given vector. A vector is written as a column: top number = horizontal movement (right is positive, left is negative), bottom number = vertical movement (up is positive, down is negative).

What to state when describing a translation: the translation vector.

Worked example 3 — translate by vector (3, −2)

A point at (1, 5) translated by the vector (3, −2):

New position: (1 + 3, 5 + (−2)) = (4, 3)

The shape moves 3 units right and 2 units down. Its size and orientation remain unchanged.

4. Enlargement

An enlargement changes the size of a shape by a scale factor from a centre of enlargement. The shape remains the same proportions (it is similar to the original).

What to state when describing an enlargement: scale factor and centre of enlargement.

Scale factor > 1: shape gets bigger.

Scale factor between 0 and 1 (a fraction): shape gets smaller.

Scale factor of 2: all lengths double; area multiplies by 4.

Worked example 4 — enlargement with scale factor 3, centre (0, 0)

Point A at (2, 1) enlarged by scale factor 3 from the origin:

A' = (2 × 3, 1 × 3) = (6, 3)

Point B at (4, 1): B' = (12, 3). Point C at (2, 5): C' = (6, 15).

Summary table — the four transformations

Transformation | What changes | What stays the same | Key information needed
Reflection | Position, orientation | Size, shape | Mirror line
Rotation | Position, orientation | Size, shape | Angle, direction, centre
Translation | Position | Size, shape, orientation | Vector
Enlargement | Size, position | Shape (angles), orientation | Scale factor, centre

Transformations in the national curriculum

The DfE's KS3 mathematics programme of study requires pupils to identify and describe the properties of symmetry of 2-D shapes, and to apply the four transformations — including reflections in the axes and lines parallel to them, rotations, translations, and enlargements. BBC Bitesize's KS3 resources confirm that describing transformations fully (with all required information) is a common mark-losing point at GCSE.

Common mistakes with transformations

Mistake | Example | How to avoid it
Reflecting in the wrong axis | Reflecting in the x-axis instead of the y-axis | Read the mirror line carefully; sketch it first
Missing the direction in a rotation | Writing "90° about (0, 0)" without clockwise/anticlockwise | Always state direction; both clockwise and anticlockwise are valid at 90°
Incorrect translation vector (sign error) | Moving right instead of left | Positive x = right, negative x = left; positive y = up, negative y = down
Not giving the centre in an enlargement | "Scale factor 2" without the centre | The centre of enlargement determines WHERE the image appears

Frequently asked questions

What information do I need to fully describe a rotation?

You need three things: (1) the angle of rotation (e.g. 90°, 180°), (2) the direction (clockwise or anticlockwise — not needed for 180° as the result is the same), and (3) the centre of rotation (a coordinate, e.g. the origin or a specific point). Missing any one of these loses marks in an exam.

What is the order of rotational symmetry?

The order of rotational symmetry is the number of times a shape looks identical to its original position during one full 360° turn. A square has order 4 (it looks the same at 90°, 180°, 270°, and 360°). An equilateral triangle has order 3. If a shape only looks the same after a full 360° turn, its order is 1, which means it has no rotational symmetry.

How is an enlargement different from the other three transformations?

Reflection, rotation, and translation are all congruent transformations — the image is exactly the same size and shape as the original. Enlargement is a similarity transformation — the image is the same shape (angles are preserved) but a different size. The ratio of corresponding sides equals the scale factor.

How do I find where a reflection lands when the mirror line is not an axis?

Draw the mirror line on the grid. For each vertex of the shape, draw a line perpendicular to the mirror line and extend it an equal distance on the other side. Plot the new vertex at that point. The perpendicular distance from the vertex to the mirror line equals the perpendicular distance from the image vertex to the mirror line.

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