RD Sharma Class 7 Maths Chapter 18 Exercise 18.3: Determine the order of rotational symmetry for each figure when rotated about the marked point (x).

Let's solve the problem step by step and determine the order of rotational symmetry for each of the given figures when rotated about the marked point (X). The order of rotational symmetry is the number of times a figure fits onto itself during a full rotation of 360°.

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🔷 What is Rotational Symmetry?

A figure has rotational symmetry if it looks exactly the same after being rotated around a central point by a certain angle. The order of rotational symmetry is the number of times this happens in a full turn (360°).

For example:
- If a shape matches itself 4 times during a full rotation, its order is 4.
- If it matches 3 times, order is 3, etc.

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Now let’s analyze each figure:

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✔ (i) Cross Shape (Plus Sign)

- It has four arms: up, down, left, right.
- When rotated by 90°, it looks the same.
- After 90°, 180°, 270°, and 360°, it matches itself → 4 positions.
- So, order = 4

> ✔️ Answer: 4

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✔ (ii) Three Arms at 120° Angles

- There are three arms equally spaced at 120° apart.
- Rotating by 120°, 240°, and 360° will make it look the same.
- So, it matches itself 3 times in 360°.
- Order = 3

> ✔️ Answer: 3

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✔ (iii) Circle Divided into 3 Equal Parts

- Similar to (ii), it has 3 equal sectors.
- Rotating by 120°, 240°, and 360° gives same appearance.
- So, order = 3

> ✔️ Answer: 3

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✔ (iv) Windmill-like Figure (4 Curved Blades)

- Has 4 identical curved blades arranged symmetrically.
- Rotates every 90° → matches at 90°, 180°, 270°, 360°.
- So, 4 times → order = 4

> ✔️ Answer: 4

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✔ (v) Straight Line with Arrow

- A single line segment with an arrowhead.
- Only matches itself when rotated by 360° (no other rotation will preserve direction).
- But note: if we ignore direction, it might match at 180° — but since there's an arrow, direction matters.
- At 180°, the arrow points opposite → not the same.
- So only matches at 360° → only once.
- But wait: does it have rotational symmetry?

Actually, a straight line segment without arrows would have rotational symmetry of order 2 (at 180°). But here, there is an arrow, so it has no rotational symmetry except at 360°.

So, it does not fit onto itself more than once → order = 1

> ✘ But some may argue: if you rotate 180°, it still looks like the same shape, just reversed. However, since the arrowhead is directional, it doesn't map to itself.

Hence, only at 360° → order = 1

> ✔️ Answer: 1

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✔ (vi) Pinwheel with 4 Triangles

- Four identical triangles arranged like a pinwheel.
- Rotates every 90° → matches at 90°, 180°, 270°, 360°.
- So, order = 4

> ✔️ Answer: 4

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✔ (vii) Regular Pentagon

- A regular pentagon has 5 equal sides and angles.
- Rotational symmetry occurs every 72° (since 360° ÷ 5 = 72°).
- Matches at 72°, 144°, 216°, 288°, 360° → 5 times
- So, order = 5

> ✔️ Answer: 5

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✔ (viii) Regular Star (Five-Pointed Star)

- This is a regular star polygon (like a pentagram).
- It has 5 points.
- Rotational symmetry every 72° → matches at 72°, 144°, 216°, 288°, 360°
- So, order = 5

> ✔️ Answer: 5

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✔ (ix) Three-Legged Figure (Like a Triskelion)

- Looks like three bent arms or legs rotating around center.
- Each arm is identical and separated by 120°.
- Rotates every 120° → matches at 120°, 240°, 360° → 3 times
- So, order = 3

> ✔️ Answer: 3

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✔ Final Answers:

| Figure | Order of Rotational Symmetry |
|-------|-------------------------------|
| (i) | 4 |
| (ii) | 3 |
| (iii) | 3 |
| (iv) | 4 |
| (v) | 1 |
| (vi) | 4 |
| (vii) | 5 |
| (viii)| 5 |
| (ix) | 3 |

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📝 Summary Explanation:

The order of rotational symmetry is determined by how many times a figure maps onto itself during a complete 360° rotation around the marked point.

- For shapes with n-fold symmetry, the rotation angle is $ \frac{360^\circ}{n} $.
- We count how many such positions exist where the figure looks identical.

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✔ Final Answer:

(i) 4, (ii) 3, (iii) 3, (iv) 4, (v) 1, (vi) 4, (vii) 5, (viii) 5, (ix) 3

Let me know if you'd like a diagram explanation or visual aid!