Symmetry Chapter Notes | Maths for Class 6 (Ganita Prakash) - New NCERT PDF Download

Introduction

Symmetry is a concept that we often see around us. It refers to the balance and harmony in shapes and objects. When parts of a shape or object are arranged in a balanced way, we say the object has symmetry.

For example, think about a flower. When you look at it from different angles, it often looks the same. This is because the petals are arranged in a way that repeats around the center. Similarly, a butterfly is not only beautiful because of its colors but also because its wings are usually mirror images of each other, creating a symmetrical pattern.

Line of Symmetry

If you fold a shape and the two halves do not match, is it a line of symmetry?

No, it is not a line of symmetry.

When we fold a shape along a certain line and both halves match exactly, that line is called a line of symmetry. 

For instance, in a blue triangle, if we draw a dotted line down the middle and fold it, the two sides cover each other perfectly. This shows that the dotted line is a line of symmetry for the triangle.

However, if we look at a shape like four puzzle pieces with a dotted line through the middle, folding it would not make the two sides match perfectly. This means that the line is not a line of symmetry for that shape.

Figures with more than one line of symmetry

Some shapes, like a square or a circle, have more than one line of symmetry. This means there are several ways to fold them so that the two halves are identical.

Example:

• Square: You can fold it four ways to see all the lines of symmetry.
  
• Rectangle: It has only two lines of symmetry, one vertical and one horizontal.
  

Reflection

When a shape has a line of symmetry, one part of the shape is a mirror image of the other. This is called reflection symmetry. It’s like looking at yourself in a mirror—what you see is a reflection of your real self.

Example: Butterfly: Its left wing is a mirror image of its right wing.

Generating Shapes Having Lines of Symmetry

Creating symmetrical shapes is a fun and creative way to explore the concept of symmetry. There are several methods to generate such shapes, and two popular techniques are using ink blots and paper folding and cutting. Let's explore how these methods work.

1. Ink Blot Devils: This is a simple and exciting method to create symmetrical shapes using ink or paint.
Steps:

1. Fold a Paper: Take a clean sheet of paper and fold it in half.
   
2. Apply Ink or Paint: Open the paper and put a few drops of ink or paint on one half of the paper.
3. Press and Open: Fold the paper back and press it so that the ink spreads. Then, carefully open the paper to reveal the design.

2. Paper Folding and Cutting: Another way to create symmetrical designs is by folding and cutting paper. This technique can produce intricate and beautiful patterns.
Steps:

1. Fold the Paper: Take a sheet of paper and fold it along the dotted line (this line can be any straight line across the paper).
   
2. Cut Along the Fold: While the paper is folded, cut along the dotted line as shown in the figure.
3. Unfold to Reveal: After cutting, carefully unfold the paper to reveal the pattern.
   

Rotational Symmetry

Does a circle have rotational symmetry?

Yes, a circle has infinite rotational symmetry because it looks the same at any angle.

Some shapes look the same even after being rotated (turned around a point). This is called rotational symmetry. For example, a square looks the same when you rotate it by 90°, 180°, 270°, or 360°.

Examples:

• The paper windmill in the picture has rotational symmetry because it looks the same when rotated by 90 degrees (a quarter turn) around the red point at its center.
  
• Angles of symmetry for the windmill are 90 degrees, 180 degrees, 270 degrees, and 360 degrees. When rotated by 360 degrees, any figure returns to its original position, making it a universal angle of symmetry.
• Other shapes, like a square, also have multiple angles of symmetry. A square overlaps with itself after rotating 90 degrees.
  
• For example, when rotating a strip shape, it does not return to its original position until a full 360-degree rotation, indicating it lacks rotational symmetry.
  
• Shapes like the windmill and square help us understand how rotational symmetry works with different angles and fixed points.

Rotational Symmetry of Figures with Radial Arms

When a shape has radial arms (like the blades of a fan), it can have rotational symmetry. The number of arms often tells us how many times the shape looks the same during one full rotation.

Example: Fan with four blades: It has four angles of symmetry—90°, 180°, 270°, and 360°.

• The figure with 4 radial arms has 4 angles of symmetry. The angles of symmetry are at 90° intervals around the center.
• If you change the angles between the radial arms, you can still have 4 angles of symmetry as long as the total of the angles is 360° and they are evenly spaced.
  
• To create a figure with only 2 angles of symmetry, you can modify the angles between the radial arms so that they are not evenly spaced. For example, having one angle much larger than the others can achieve this.
• With 3 radial arms, the figure has rotational symmetry only with a full rotation of 360°. To have 3 angles of symmetry, the angles between the radial arms must be equal, specifically 120° each.
  
• To achieve this, the angles between the dotted lines must be 120° each, which allows the figure to overlap with its rotated versions.
  
• The figure with 3 arms will have the following angles of rotation: 120°, 240°, 360°
  
• These angles correspond to rotating the figure by 120° or 240°, and the full rotation of 360° brings it back to the original position.

Symmetries of a Circle

A circle is a unique shape because it has infinite lines of symmetry. You can draw a line from the center to any point on the edge, and it will divide the circle into two identical parts. Similarly, rotating a circle by any angle keeps it looking the same.

Example: Wheel: No matter how you turn it, it always looks the same.

Key Words

• Symmetry: A figure has symmetry if parts of it repeat in a definite pattern.
• Line of Symmetry: A line that divides a shape into identical parts.
• Reflection Symmetry: One part of the shape is a mirror image of the other.
• Rotational Symmetry: A shape looks the same after a certain rotation.
• Circle Symmetry: A circle has infinite lines of symmetry and can be rotated by any angle without changing its appearance.