11.1 Symmetry & Transformations

Symmetry questions appear from Paper B onwards, and transformation questions run right through to Papers G-H. ICAS loves this topic because it tests careful observation rather than calculation: a typical item shows two shapes on a grid and asks which single transformation maps one onto the other, or asks you to count lines of symmetry in an unfamiliar logo. Because you cannot bring a calculator, everything here is done by eye, by counting grid squares, and by reasoning — which is exactly how you should practise.

Line Symmetry

A shape has line symmetry (also called mirror symmetry or reflection symmetry) if you can draw a line through it so that one half is the mirror image of the other. The line itself is called an axis of symmetry.

Key facts worth memorising:

Notice the pattern: a regular polygon with n sides has n lines of symmetry. The common trap is the rectangle and the parallelogram. Students often claim a rectangle has 4 lines of symmetry, but the diagonals are NOT axes — fold a rectangle along a diagonal and the corners do not match. A parallelogram has no line symmetry at all, which surprises many students.

Rotational Symmetry

A shape has rotational symmetry if it fits onto itself (looks identical) when rotated through less than one full turn about its centre. The order of rotational symmetry is the number of times the shape fits onto itself during a complete 360° turn.

• Equilateral triangle: order 3 (fits at 120°, 240°, 360°)
• Square: order 4 (every 90°)
• Rectangle and parallelogram: order 2 (every 180°)
• Regular hexagon: order 6
• Every shape has order at least 1, because anything fits onto itself after a full turn

A quick formula: for a shape of order n, the smallest rotation angle is 360° ÷ n. ICAS phrases these as: "What is the order of rotational symmetry of this figure?" — watch for shapes like a swastika-style pinwheel or the recycling symbol, which have rotational symmetry (order 2 or 3) but no line symmetry.

The Four Transformations

Reflection

A reflection flips a shape over a mirror line. Every point and its image are the same perpendicular distance from the mirror line, on opposite sides. Reflection changes the orientation of a shape (a clockwise-labelled shape becomes anticlockwise) but keeps size and shape identical.

Worked example: reflect the point (3, 2) in the vertical line x = 5. The point is 2 units left of the mirror line, so its image is 2 units right of it: (7, 2). For a diagonal mirror line such as y = x, swap the coordinates: (3, 2) reflects to (2, 3).

Translation

A translation slides every point by the same distance in the same direction. Describe it with two movements, e.g. "3 units right and 2 units up". Size, shape AND orientation all stay the same — the image is never flipped or turned. On a grid, count squares from one vertex to its image; every vertex moves by exactly the same amount.

Rotation

A rotation turns a shape about a fixed point called the centre of rotation. You must state three things: the centre, the angle, and the direction. "90° clockwise about the origin" is complete; "90°" alone is not. Orientation is preserved (the shape is turned, not flipped). Useful moves on a grid about the origin: 90° clockwise sends (a, b) to (b, −a); 180° sends (a, b) to (−a, −b).

Enlargement

An enlargement changes the size of a shape but keeps its shape — the image is similar to the original, with all angles unchanged. The scale factor k multiplies every length: scale factor 3 means each side is 3 times longer. A scale factor between 0 and 1 (say, ½) is still called an enlargement even though it makes the shape smaller. Lengths multiply by k, but areas multiply by k²: a triangle enlarged with scale factor 2 has sides twice as long but an area four times as big. This length-versus-area distinction is one of ICAS's favourite traps for Papers G-H.

Identifying Which Transformation Maps One Shape onto Another

Use this decision checklist on grid questions:

1. Different sizes? It must be an enlargement (or a combination including one).
2. Same size, but flipped (orientation reversed)? It is a reflection. Find the mirror line halfway between corresponding points.
3. Same size, same orientation, just slid? Translation. Count the horizontal and vertical shift of any vertex.
4. Same size, same orientation, but turned? Rotation. Try 90°, 180° or 270° about likely centres (the origin, or a shared vertex) until every vertex matches.

Worked example: triangle P has vertices (1, 1), (4, 1), (1, 3). Triangle Q has vertices (−1, 1), (−4, 1), (−1, 3). Same size, and the shapes are mirror images — orientation is reversed — so this is a reflection. Each pair of corresponding points is the same distance from the y-axis, so the mirror line is x = 0 (the y-axis).

Combining Transformations and Tessellations

ICAS senior papers sometimes describe two transformations in a row, e.g. "reflect in the x-axis, then translate 2 units up". Do them one at a time, point by point. Note that order matters: reflecting then translating usually gives a different result from translating then reflecting.

A tessellation is a pattern of shapes that covers a flat surface with no gaps and no overlaps. Every triangle and every quadrilateral tessellates. Of the regular polygons, only the equilateral triangle, square and regular hexagon tessellate on their own, because their interior angles (60°, 90°, 120°) divide exactly into 360°. Regular pentagons (108°) do not. Semi-regular tessellations mix two or more regular polygons — octagons with squares is the classic example.