This MCQ module is based on: Combining Line and Rotational Symmetry
TOPIC 35 OF 41
Combining Line and Rotational Symmetry
🎓 Class 6
Mathematics
CBSE
Theory
Ch 9 — Symmetry
⏱ ~30 min
🌐 Language: [gtranslate]
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This mathematics assessment will be based on: Combining Line and Rotational Symmetry
Targeting Class 6 level in Geometry, with Basic difficulty.
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9.5 When a Figure Has Both Types of Symmetry
Many figures show both line and rotational symmetry. A square, a regular hexagon, and a circle all do. Others — like the letter S — show only rotational symmetry. Some — like the letter A — show only line symmetry. And some — like the letter F — show neither.
Four possibilities
Every figure falls into exactly one of four groups:① Line symmetry only (e.g. letter A, isosceles triangle) ② Rotational only (e.g. letter S, pinwheel) ③ Both (e.g. square, circle, regular hexagon) ④ Neither (e.g. letter F, scalene triangle)
Four types of figures, based on which symmetries they have.
A Useful Pattern
For regular polygons (all sides equal, all angles equal), the number of lines of symmetry equals the order of rotational symmetry — both equal the number of sides.
| Regular Polygon | Lines of Symmetry | Order of Rotation | Angle of Rotation |
|---|---|---|---|
| Equilateral triangle (3-sides) | 3 | 3 | 120° |
| Square (4-sides) | 4 | 4 | 90° |
| Regular pentagon (5) | 5 | 5 | 72° |
| Regular hexagon (6) | 6 | 6 | 60° |
| Regular n-gon | \(n\) | \(n\) | \(360°/n\) |
| Circle | ∞ | ∞ | any |
Example — Rangoli Pattern Analysis
Consider an 8-petalled rangoli. Each petal is identical. Rotating by 45° maps each petal onto the next — so order = 8. It also has 8 lines of symmetry (through each petal and between each pair). Hence: regular 8-fold symmetry.
An 8-petalled rangoli with 8-fold rotational + line symmetry.
Symmetry in Digits & Letters
Look at the digit 8. It has two lines of symmetry (horizontal and vertical) and rotational symmetry of order 2. The digit 0 (drawn as an oval) has 2 lines and order 2. The digit 6 rotated by 180° becomes 9 — this pair shows rotational partnership but not self-symmetry.
🔵 Investigate: Which digits 0-9 have both line and rotational symmetry? 0 and 8 (both have 2 lines + order 2). 1, 3 have line symmetry only. 2, 4, 5, 6, 7, 9 have neither (in standard printed form).
Activity: Classify the Classroom
L4 AnalyseMaterials: Paper, pencil, classroom objects
Predict: Of all the objects you can see right now, guess what fraction show both line and rotational symmetry.
- Make a 4-column table: Line only / Rotational only / Both / Neither.
- Pick 12 objects from your surroundings: classroom items, stationery, designs on clothes, tiles, jewellery, etc.
- For each object, check (a) does it have at least one line of symmetry? (b) does it have rotational symmetry of order ≥ 2?
- Record each object under the correct column.
- Discuss with a partner whether any of your classifications are borderline or debatable.
Typical finds: Both — ceiling fan (if blades identical), wall clock face, many tiles, window grills, coins. Line only — chair, school bag, many alphabets on posters. Rotational only — some spiral logos, pinwheels with odd tilt. Neither — crumpled paper, school notebooks with only a title.
Worked Example — A Letter Analysis
Analyse the letter H. It has 2 lines of symmetry — one vertical, one horizontal. Rotating it by 180° about its centre gives back the original. So order = 2. Thus H has both types of symmetry.
Analyse the letter N. It has no line of symmetry. But rotating it by 180° gives back an N — so order = 2 (rotational only).
Figure it Out — Part C
Q1. Classify each of the following under "Line only / Rotational only / Both / Neither":
(a) Equilateral triangle (b) Letter N (c) Parallelogram (non-rectangular) (d) Scalene triangle (e) Regular hexagon (f) Letter B.
(a) Equilateral triangle (b) Letter N (c) Parallelogram (non-rectangular) (d) Scalene triangle (e) Regular hexagon (f) Letter B.
(a) Both — 3 lines + order 3. (b) Rotational only — order 2. (c) Rotational only — order 2, no line symmetry. (d) Neither. (e) Both — 6 lines + order 6. (f) Line only — 1 horizontal line, order 1.
Q2. A figure has 4 lines of symmetry. Must it also have rotational symmetry? What is its minimum order?
Yes. Two lines of symmetry at angle θ imply a rotational symmetry of angle 2θ. With 4 equally-spaced axes, we get rotation of 90° — minimum order 4 (as in a square).
Q3. Draw a figure with rotational symmetry of order 3 but no line of symmetry. Describe your construction.
Start with an equilateral triangle. On one side, add a small "comma" shape pointing in the same rotational direction on each side (say clockwise). Each 120° rotation matches the next comma, but because each comma is curved in one direction, no line-fold makes it match itself. This is called a triskele pattern.
Competency-Based Questions
Scenario: Ravi's art teacher asks the class to design a 20 cm × 20 cm tile so that when four tiles are placed together, the combined pattern has both 4-fold rotational symmetry and 4 lines of symmetry. Ravi draws a single diagonal line from one corner of his tile to the opposite corner.
Q1. What symmetry does Ravi's single tile have?
L3 ApplyRavi's tile has only 1 line of symmetry (the drawn diagonal itself) and no rotational symmetry (order 1). The other diagonal is not a line because only one diagonal is drawn.
Q2. Analyse whether placing 4 copies of Ravi's tile together can create the required 4-fold symmetric pattern.
L4 AnalyseYes — if the 4 tiles are rotated 0°, 90°, 180°, 270° and arranged into a 2 × 2 grid sharing a common centre, the 4 diagonals together form a plus (+) and an X — producing order 4 and 4 axes (both diagonals + both midlines) for the combined 40 × 40 tile.
Q3. Evaluate which tile arrangement gives more lines of symmetry — (i) four identical tiles placed the same way, or (ii) four tiles rotated as in Q2.
L5 Evaluate(i) All four tiles placed the same way → pattern has only 1 line of symmetry (matching the original single diagonal direction). (ii) Rotated arrangement → 4 lines. So (ii) gives more lines. Rotation breaks the repetition and introduces new axes.
Q4. Create a tile pattern where 4 copies placed together form a design with rotational order 4 but zero lines of symmetry. Describe it.
L6 CreateOn a single tile, draw a small curved comma (or spiral) pointing clockwise from centre to corner. Arrange 4 such tiles rotated 0/90/180/270° — the combined motif swirls clockwise with order 4 rotation but no line symmetry (reflection reverses the swirl direction).
Assertion–Reason Questions
Assertion (A): Every regular polygon with n sides has n lines of symmetry and rotational order n.
Reason (R): All sides and angles of a regular polygon are equal.
Reason (R): All sides and angles of a regular polygon are equal.
(a) The equal-sides-and-angles condition is exactly why reflections across axes through vertices/midpoints and rotations by 360°/n all map the shape to itself.
Assertion (A): A figure with no line of symmetry cannot have rotational symmetry.
Reason (R): Line and rotational symmetry are independent properties.
Reason (R): Line and rotational symmetry are independent properties.
(d) A is false — the letter S, a pinwheel, or a parallelogram has rotational symmetry without any line symmetry. R is true and actually disproves A.
Assertion (A): The letter H has both line and rotational symmetry.
Reason (R): H has two perpendicular lines of symmetry, and intersecting lines of symmetry always imply rotational symmetry.
Reason (R): H has two perpendicular lines of symmetry, and intersecting lines of symmetry always imply rotational symmetry.
(a) Two lines meeting at angle θ produce a rotational symmetry of angle 2θ. Perpendicular lines (θ = 90°) give rotational symmetry of 180°, i.e. order 2. R correctly explains A.
Frequently Asked Questions
Which shapes have both line and rotational symmetry?
Regular polygons (equilateral triangle, square, pentagon, hexagon), circles, and many rangoli/snowflake designs have both line and rotational symmetry.
How many lines of symmetry does a regular hexagon have?
A regular hexagon has 6 lines of symmetry and rotational symmetry of order 6 (angle of rotation 60 degrees).
Does a circle have infinite symmetry?
Yes. A circle has infinitely many lines of symmetry (every diameter) and rotational symmetry of every angle about its centre.
How is symmetry used in real life?
Symmetry appears in architecture (Taj Mahal), art (rangoli, mandala), nature (butterflies, flowers), logos and traffic signs for balance and beauty.
Does a figure with rotational symmetry always have line symmetry?
No. A pinwheel and the letter S have rotational symmetry of order 2 but no line of symmetry. Rotational and line symmetry are independent properties.
What is dihedral symmetry?
Dihedral symmetry is the combination of rotational and reflective (line) symmetry found in regular polygons and many decorative patterns.
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Mathematics Class 6 — Ganita Prakash
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