Symmetry
1. What is Symmetry?
A shape has SYMMETRY if it can be transformed (flipped, turned, or slid) so that it looks the SAME as before.
'Nature loves symmetry — butterflies, snowflakes, flowers, and even our own bodies show symmetry.'
Types of Symmetry
| Type | Description |
|---|---|
| Line symmetry (Reflection) | Shape can be DIVIDED by a line into two mirror-image halves |
| Rotational symmetry | Shape can be ROTATED (less than 360°) to look the same |
| Point symmetry | Shape looks the same when rotated by 180° about a point |
2. Line Symmetry (Reflection Symmetry)
A line of symmetry divides a figure into TWO IDENTICAL halves that are MIRROR IMAGES of each other.
| Shape | Number of Lines of Symmetry |
|---|---|
| Scalene triangle | 0 |
| Isosceles triangle | 1 |
| Equilateral triangle | 3 |
| Square | 4 |
| Rectangle | 2 |
| Rhombus | 2 |
| Circle | INFINITE |
| Regular pentagon | 5 |
| Regular hexagon | 6 |
| Parallelogram (non-rectangle) | 0 |
Letters with line symmetry:
- Vertical line: A, H, I, M, O, T, U, V, W, X, Y
- Horizontal line: B, C, D, E, H, I, K, O, X
- Both: H, I, O, X
3. Rotational Symmetry
A shape has rotational symmetry if it looks IDENTICAL to its original position after a rotation (of less than 360°) about its CENTRE.
Key Terms
- Centre of rotation: The FIXED point about which the shape rotates
- Angle of rotation: The SMALLEST angle through which the shape rotates to look the same
- Order of rotational symmetry: The NUMBER of times a shape looks the SAME in one full 360° rotation
Order = 360° / Angle of rotation
4. Examples of Rotational Symmetry
| Shape | Order | Angle of Rotation |
|---|---|---|
| Equilateral triangle | 3 | 120° |
| Square | 4 | 90° |
| Rectangle | 2 | 180° |
| Rhombus | 2 | 180° |
| Regular pentagon | 5 | 72° |
| Regular hexagon | 6 | 60° |
| Circle | INFINITE | Any angle |
| Parallelogram | 2 | 180° |
| Scalene triangle | 1 | 360° (no rotational symmetry) |
| Letter 'S' | 2 | 180° |
Worked Example: A regular octagon has rotational symmetry of order 8. Find its angle of rotation.
Angle = 360°/8 = 45°
5. Point Symmetry
A shape has point symmetry if every point has a MATCHING point at the SAME distance from the centre but in the OPPOSITE direction.
'Point symmetry is the SAME as rotational symmetry of order 2 (180° rotation).'
Shapes with point symmetry:
- All parallelograms (including rectangle, rhombus, square)
- Regular hexagon
- Letter N, S, Z, H, I, O, X
6. Reflection and Mirror Images
When an object is reflected in a mirror:
- The image is the SAME SIZE as the object
- The image is LATERALLY INVERTED (left ↔ right swapped)
- The LINE OF REFLECTION is the perpendicular bisector of the segment joining a point to its image
Worked Example: What is the reflection of 'ICSE' in a vertical mirror?
ICSE reflected in a vertical mirror: The letters reverse left-to-right.
Common Mistakes and Fixes
| Mistake | Fix |
|---|---|
| 'A parallelogram has no lines of symmetry' | Correct — a non-rectangle parallelogram has NO lines of symmetry |
| 'A rectangle has 4 lines of symmetry' | A rectangle has 2 (the diagonals are NOT lines of symmetry) |
| 'Rotational symmetry of order 1 means no rotational symmetry' | Order 1 means it looks the same only after 360° — TECHNICALLY no rotational symmetry |
| 'All triangles have line symmetry' | Only isosceles and equilateral triangles have line symmetry |
ICSE Exam Focus (4–5 marks)
- 2-mark questions: Identify lines of symmetry in shapes/letters
- 3-mark questions: Find order and angle of rotational symmetry
- 4-mark questions: Complete a figure given its line(s) of symmetry
- 5-mark questions: Combined — line and rotational symmetry analysis
Self-Test
Q1. How many lines of symmetry does an equilateral triangle have? A1. THREE — one from each vertex to the midpoint of the opposite side.
Q2. What is the order of rotational symmetry of a rectangle? A2. Order 2 (looks the same after 180° and 360° rotation).
Q3. Does the letter 'Z' have point symmetry? Explain. A3. Yes, 'Z' has point symmetry (rotational symmetry of order 2). Turned 180°, it looks the same.
Q4. A shape has rotational symmetry of order 8. What is its angle of rotation? A4. Angle = 360°/8 = 45°.
Q5. How many lines of symmetry does a regular hexagon have? A5. SIX — three through opposite vertices and three through midpoints of opposite sides.
Q6. Does a parallelogram (non-rectangle, non-rhombus) have line symmetry? Does it have rotational symmetry? A6. A general parallelogram has NO line symmetry. It has rotational symmetry of ORDER 2 (180° rotation).
