Find reflections and rotations that preserve a shape
Identify a mirror line by matching the whole figure · State rotational order and smallest positive symmetry angle · Distinguish generic shapes from special cases
Section Paper 3 · Historical Core · calculator
Exam scope and assessment notes
Core and Extended foundations. Historical study of Cambridge IGCSE Mathematics (US) 0444, 2025 Version 2. The final assessment was June 2025; the syllabus was available only to schools in Arizona, USA. This is not a current exam-entry option. Original self-check teaching and explained practice are not Cambridge papers, official marks or predicted grades. Linked teaching foundations do not establish full syllabus mastery. Calibrated cumulative assessment, reliable drawn-work review and qualified mathematics-teacher review remain unfinished.
Before you begin
- Plane shape definitions
- Degrees in a full turn
Work on paper. Try each question before revealing help, and explain why your method works.
Quick readiness check
What should you establish before applying the method in this lesson?
What you will learn
- Identify a mirror line by matching the whole figure
- State rotational order and smallest positive symmetry angle
- Distinguish generic shapes from special cases
Mirror and rotation tests
A line of symmetry reflects the entire figure onto itself; it is not just a line dividing the area equally. Rotational order counts the matches in a full 360° turn, including the return to the starting position. For a finite order n, the smallest positive matching rotation is 360°/n.
A square has four mirror lines and order 4; a non-square rectangle has two mirror lines and order 2. Its diagonals generally do not preserve it under reflection. A generic parallelogram that is neither a rectangle nor a rhombus has no mirror line and order 2.
State the shape assumptions
A regular n-sided polygon has n mirror lines and order n. An equilateral triangle has three mirror lines and order 3; an isosceles triangle that is not equilateral has one mirror line and order 1. A circle is preserved by reflection in any diameter and by any rotation about its center, so no smallest positive symmetry rotation exists.
Sources for this lesson
Pause and explain
Are a non-square rectangle's diagonals mirror lines?
Worked example
Compare a square with a non-square rectangle.
Show the worked solution
- Square: four mirror lines, order 4, smallest rotation 90°.
- Non-square rectangle: two mirror lines, order 2, smallest rotation 180°.
Answer Square 4/4/90°; non-square rectangle 2/2/180°.
Try a topic test
Choose Easy, Normal or Difficult. Save your answers and open worked review after finishing.
Test this topic →From guided practice to a new situation
Write your answer and reasoning first. Use a hint only if you are stuck. The model solution is for self-checking; your written work is not automatically marked.
Give the mirror-line count, rotational order and smallest matching rotation of a regular hexagon.
Give me a hint
Use n = 6.
Compare my reasoning
- It has six mirror lines and order 6.
- 360°/6 = 60°.
6 mirror lines; order 6; 60°.
Look for these in your work
- Use regularity
- Count the full-turn match
Compare a non-equilateral isosceles triangle with an equilateral triangle.
Give me a hint
Keep the non-equilateral condition.
Compare my reasoning
- The first has one mirror line and rotational order 1.
- The second has three mirror lines and order 3, with a 120° smallest rotation.
Isosceles non-equilateral: 1 mirror, order 1. Equilateral: 3 mirrors, order 3, 120°.
Look for these in your work
- Respect the special case
- Separate reflection and rotation
A design is a generic parallelogram, neither rectangular nor rhombic. Does a diagonal form a mirror line? What rotation preserves it?
Give me a hint
Check the full shape rather than equal areas.
Compare my reasoning
- It has no mirror lines, including its diagonals.
- A 180° rotation about the diagonal intersection matches it.
No mirror line; rotational order 2, preserved by 180°.
Look for these in your work
- Reject an area-only argument
- State the rotation center
Common mistakes
- An equal-area divider need not be a mirror line.
- Special cases can have extra symmetry.
What can you explain now?
Are a non-square rectangle's diagonals mirror lines?
Compare with the explanation
No; reflecting in them does not match the whole rectangle.
Only a square has those additional diagonal mirror symmetries.
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Make it stick
Are a non-square rectangle's diagonals mirror lines?
If you needed the solution, close it and solve the problem again from a blank page. A correct answer is stronger when you can explain the reason for each step.