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Symmetries of a Circle

A circle has reflection symmetry about every diameter and rotational symmetry about its centre for any angle of rotation.

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Student-friendly explanation

If a circle is folded along any diameter, the two halves match exactly. If it is rotated around its centre, it still looks the same because all boundary points remain at the same distance from the centre.

How to write this in exams

  1. 1

    Start with the exact idea

    A circle has reflection symmetry about every diameter and rotational symmetry about its centre for any angle of rotation.

  2. 2

    Then show how to use it

    Look for the centre. Check if the proposed mirror line passes through it. For rotation, check whether the rotation is about the centre, not about a point on the circle.

  3. 3

    Add one concrete example

    A wheel rotated by 40 degrees about its centre still has the same circular outline. A fold along any line through the centre divides the circle into two equal halves.

  4. 4

    Avoid this incomplete answer

    Saying a circle has only two lines of symmetry is wrong because infinitely many diameters can be drawn through the centre.

Definition

A circle has reflection symmetry about every diameter and rotational symmetry about its centre for any angle of rotation.

Example

A wheel rotated by 40 degrees about its centre still has the same circular outline. A fold along any line through the centre divides the circle into two equal halves.

Rule to remember

Reflection symmetry: every diameter is an axis of symmetry. Rotational symmetry: a circle matches itself after rotation about the centre by any angle.

Memory hook

Through the centre, perfect mirror.

Examples and method

Worked example

If a line l cuts a circle into two mirror-image halves and passes through centre O, l is a diameter and an axis of symmetry.

Method to apply

Look for the centre. Check if the proposed mirror line passes through it. For rotation, check whether the rotation is about the centre, not about a point on the circle.

Diagram support

Draw several diameters through the same centre and show each one splitting the circle into equal halves.

How CBSE asks it

Students may be asked to identify lines of symmetry, rotational symmetry, or explain why a given line is not a symmetry line.

Avoid common mistakes

Common confusion

Students sometimes think only horizontal and vertical diameters are lines of symmetry. Actually, every diameter is a line of symmetry.

Common wrong answer

Saying a circle has only two lines of symmetry is wrong because infinitely many diameters can be drawn through the centre.

Exam tip

For symmetry questions, check whether the line passes through the centre. A chord not passing through the centre is not a line of symmetry of the whole circle.

Quick check

Why is any diameter a line of symmetry of a circle?

Any diameter is a line of symmetry because it passes through the centre and divides the circle into two matching semicircles.

Answer writing and exam use

1-mark answer

A circle has reflection symmetry about every diameter and rotational symmetry about its centre for any angle of rotation.

2-mark answer

A circle has reflection symmetry about every diameter and rotational symmetry about its centre for any angle of rotation. Reflection symmetry: every diameter is an axis of symmetry. Rotational symmetry: a circle matches itself after rotation about the centre by any angle. A wheel rotated by 40 degrees about its centre still has the same circular outline. A fold along any line through the centre divides the circle into two equal halves.

3-mark answer

If a circle is folded along any diameter, the two halves match exactly. If it is rotated around its centre, it still looks the same because all boundary points remain at the same distance from the centre. Reflection symmetry: every diameter is an axis of symmetry. Rotational symmetry: a circle matches itself after rotation about the centre by any angle. If a line l cuts a circle into two mirror-image halves and passes through centre O, l is a diameter and an axis of symmetry. Students may be asked to identify lines of symmetry, rotational symmetry, or explain why a given line is not a symmetry line. Saying a circle has only two lines of symmetry is wrong because infinitely many diameters can be drawn through the centre.
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