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Key Symmetry Concepts & Properties (Revision Table)
| Plane Figure | Lines of Symmetry | Centre of Rotation | Order of Rotation | Angle of Rotation |
|---|---|---|---|---|
| Equilateral Triangle | 3 | Centroid / Intersection of medians | 3 | 120° |
| Square | 4 | Intersection of diagonals | 4 | 90° |
| Rectangle | 2 | Intersection of diagonals | 2 | 180° |
| Rhombus | 2 | Intersection of diagonals | 2 | 180° |
| Parallelogram | 0 | Intersection of diagonals | 2 | 180° |
| Regular Hexagon | 6 | Center of hexagon | 6 | 60° |
| Circle | Infinitely many | Center of circle | Infinite | Any angle |
Chapter 13 Brain Teasers Solutions
Q.1 (A). Tick the correct option:
(a) The order of rotational symmetry of an equilateral triangle is:
(i) 1
(ii) 2
(iii) 3
(iv) 6
Answer: (iii) 3
Explanation: An equilateral triangle fits onto itself 3 times in one full rotation of 360° (at 120°, 240°, and 360°).
(b) The angle of rotation for a regular hexagon is:
(i) 45°
(ii) 60°
(iii) 90°
(iv) 120°
Answer: (ii) 60°
Explanation: Angle of rotation = 360° / n = 360° / 6 = 60°.
(c) Which of the following letters has rotational symmetry of order 2?
(i) A
(ii) B
(iii) N
(iv) C
Answer: (iii) N
Explanation: When the letter ‘N’ is rotated by 180°, it looks exactly identical to the original letter. Hence its order of rotational symmetry is 2.
(d) A quadrilateral having 4 lines of symmetry and rotational symmetry of order 4 is:
(i) Rhombus
(ii) Rectangle
(iii) Square
(iv) Parallelogram
Answer: (iii) Square
Explanation: A square has 4 lines of symmetry (2 along medians, 2 along diagonals) and an angle of rotation of 90° (order 4).
(e) The number of lines of symmetry in a scalene triangle is:
(i) 0
(ii) 1
(iii) 2
(iv) 3
Answer: (i) 0
Explanation: A scalene triangle has all three sides of unequal length; therefore, it cannot be folded along any line to match two halves.
Q.1 (B). Answer the following questions:
(a) Give two examples of plane figures that have rotational symmetry of order greater than 1, but NO line of symmetry.
Solution:
1. A general Parallelogram: It has rotational symmetry of order 2 (angle of rotation = 180°) but zero lines of symmetry.
2. A pinwheel / windmill with curved vanes (or Swastika symbol): It possesses rotational symmetry of order 4, but no axis of reflection symmetry.
(b) What is the angle of rotation of a regular polygon with 8 sides (regular octagon)?
Solution:
Angle of rotation = 360° / n = 360° / 8 = 45°.
Its order of rotational symmetry is 8.
(c) State the number of lines of symmetry and order of rotational symmetry for a Rhombus.
Solution:
• Number of lines of symmetry = 2 (along its two diagonals).
• Order of rotational symmetry = 2 (at 180° and 360°).
(d) Which letters of the English alphabet have both horizontal and vertical lines of symmetry?
Solution:
The English letters with both horizontal and vertical lines of symmetry are: H, I, O, and X.
(e) State whether a circle has line symmetry and rotational symmetry.
Solution:
Yes. A circle has infinitely many lines of symmetry (every diameter is a line of symmetry) and infinite order of rotational symmetry about its center (coincides with itself when rotated through any angle).
HOTS (Higher Order Thinking Skills) Questions
HOTS 1: Can a figure have rotational symmetry of order more than 1 if its angle of rotation is:
(a) 45°?
(b) 17°?
(c) 72°?
Give reasons for each case.
Solution:
For a figure to possess rotational symmetry of order greater than 1, a full turn of 360° must be an exact multiple of the angle of rotation (i.e., 360° / θ must be an integer > 1).
(a) For 45°: 360° / 45° = 8 (an integer). Yes, it has rotational symmetry of order 8.
(b) For 17°: 360° / 17° = 21.176… (not an integer). No, it cannot have rotational symmetry.
(c) For 72°: 360° / 72° = 5 (an integer). Yes, it has rotational symmetry of order 5 (like a regular pentagon).
HOTS 2: A figure is rotated about a point through an angle of 60° and coincides with its initial position. At what other angles will it also coincide with itself during a complete 360° rotation?
Solution:
If a figure coincides at 60°, it will coincide at every positive integral multiple of 60° up to 360°:
• 60° × 1 = 60°
• 60° × 2 = 120°
• 60° × 3 = 180°
• 60° × 4 = 240°
• 60° × 5 = 300°
• 60° × 6 = 360°.
Therefore, the other angles of coincidence are 120°, 180°, 240°, 300°, and 360° (Order of rotational symmetry = 6).
Value Based Questions (VBQs)
Question: During a school festival, students designed traditional floral Rangoli and Mandala art on the entrance courtyard. Priya created an 8-petal symmetric circular pattern where each petal is identical in size and shape.
(a) Find the angle of rotation and order of rotational symmetry of Priya’s 8-petal Rangoli design.
Solution:
• Order of rotational symmetry = 8.
• Angle of rotation = 360° / 8 = 45°.
(b) How many lines of symmetry does this 8-petal circular design possess?
Solution:
It possesses 8 lines of symmetry (4 passing through pairs of opposite petals, and 4 passing through mid-spaces between opposite petals).
(c) What cultural and personal values are developed by creating geometric art like Rangoli?
Solution:
1. Appreciation for cultural heritage and mathematical aesthetics.
2. Patience, precision, focus, and teamwork.
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