# Symmetry

## Overview

In this chapter, students explore the concept of symmetry through patterns, reflection, folding, rotation, and shapes in nature and architecture. It introduces the ideas of line symmetry and rotational symmetry with interactive activities like paper folding, cutting, and using ink blots to help visualize symmetric properties in figures.

## Key Topics Covered

### 1. Understanding Symmetry in Figures

* Symmetry is when parts of a figure are repeated in a definite pattern.
* Examples include butterflies, rangolis, pinwheels, and flowers.
* Objects like clouds, which lack a repeating pattern, are not symmetrical.

### 2. Line of Symmetry

* A **line of symmetry** divides a figure into two mirror halves.
* Folding a figure along this line results in the two parts overlapping exactly.
* Some figures (like squares) have multiple lines of symmetry — vertical, horizontal, and diagonal.
* Not all shapes (like rectangles) have diagonal symmetry.

### 3. Reflection Symmetry

* When one half of a figure reflects the other across the line of symmetry.
* Helps identify matching parts in folded figures.
* Example: Points A and D on a square reflect to B and C across a vertical line.

### 4. Generating Symmetric Figures

* Activities like ink blotting and paper folding help students create symmetric designs.
* Cutting folded paper can reveal symmetric shapes upon unfolding.
* Punching folded paper creates symmetric hole patterns.

### 5. Rotational Symmetry

* A figure has **rotational symmetry** if it looks the same after rotating it around a fixed point (the **centre of rotation**).
* The **angle of rotational symmetry** is the smallest angle through which the figure can be rotated and still look the same.
* Example: A windmill rotated by 90°, 180°, 270°, and 360° looks the same.
* A figure with no smaller angle of symmetry than 360° has no rotational symmetry.

### 6. Radial Arm Figures

* Figures with radial arms (like wheels or fans) help visualize rotational symmetry.
* If a figure has *n* radial arms evenly spaced, it has *n* angles of symmetry.
* For example:

* 3 arms → angles of symmetry: 120°, 240°, 360°
* 5 arms → angles of symmetry: 72°, 144°, 216°, 288°, 360°

### 7. Symmetries of a Circle

* A circle has infinite lines of symmetry (every diameter is one).
* It also has rotational symmetry at any angle about its center.

### 8. Practice Tasks and Activities

* Completing symmetric figures from partial drawings.
* Drawing triangles and quadrilaterals with specified symmetries.
* Identifying lines and angles of symmetry in designs and real-world objects like the Ashoka Chakra or Parliament Building.
* Creating symmetric tile patterns using square grids and colour tiles.

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## New Terms and Definitions

| Term | Simple Definition |
| ------------------- | ------------------------------------------------------------------------------- |
| Symmetry | A balanced and equal arrangement on both sides of a line or point |
| Line of Symmetry | A line that divides a figure into two matching parts |
| Mirror Halves | Two equal parts of a figure that match exactly when folded |
| Reflection Symmetry | When one half of a figure is a mirror image of the other |
| Rotational Symmetry | A figure looks the same after being rotated around a point |
| Centre of Rotation | The fixed point around which a figure is rotated |
| Angle of Symmetry | The smallest angle through which a shape can be rotated and still look the same |
| Radial Arms | Equal arms or extensions that spread out from a center like spokes of a wheel |
| Diagonal | A line joining two non-adjacent corners of a polygon |
| Regular Polygon | A shape with all sides and angles equal |

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**Practice Questions (with Answer and Explanation)**
**Easy (3 Questions):**

1. **Which of these shapes has only one line of symmetry?**
a. Circle
b. Square
c. Isosceles triangle
✅ **Answer:** c
📝 **Explanation:** An isosceles triangle has exactly one line of symmetry.

2. **If a square is folded along a diagonal, what do you observe?**
✅ **Answer:** Both halves overlap completely.
📝 **Explanation:** Diagonals of a square are lines of symmetry.

3. **How many lines of symmetry does a regular hexagon have?**
✅ **Answer:** 6
📝 **Explanation:** Each line passes through opposite vertices or midpoints of opposite sides.

**Medium (2 Questions):**

4. **What is the smallest angle of rotational symmetry in a regular pentagon?**
✅ **Answer:** 72°
📝 **Explanation:** 360° ÷ 5 sides = 72°.

5. **Does a rectangle have diagonal symmetry?**
✅ **Answer:** No
📝 **Explanation:** When folded along the diagonal, the two halves do not match unless it's a square.

**Difficult (3 Questions):**

6. **Find the angles of rotational symmetry for a figure with 3 radial arms.**
✅ **Answer:** 120°, 240°, 360°
📝 **Explanation:** 360° divided by 3 arms = 120° as the smallest angle.

7. **How many lines of symmetry and angles of symmetry does the Ashoka Chakra have?**
✅ **Answer:** 24 lines and 24 angles of symmetry
📝 **Explanation:** It has 24 spokes (radial arms).

8. **Can a figure have reflection symmetry but not rotational symmetry? Give an example.**
✅ **Answer:** Yes, a figure like an isosceles triangle has reflection but no rotational symmetry except 360°.

**Very Difficult (2 Questions):**

9. **Is it possible to draw a triangle with exactly 2 lines of symmetry?**
✅ **Answer:** No
📝 **Explanation:** A triangle can have either 0, 1, or 3 lines of symmetry — not 2.

10. **A figure has a smallest angle of symmetry of 60°. List all its angles of symmetry.**
✅ **Answer:** 60°, 120°, 180°, 240°, 300°, 360°
📝 **Explanation:** All are multiples of 60°, up to 360°.

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