12.1 Introduction

1. Importance of Symmetry:

• Symmetry is a fundamental geometrical concept that appears frequently in nature and various human activities.
• It’s utilized by artists, designers, car manufacturers, architects, and many other professionals.
• Examples of symmetry can be seen in beehives, flowers, tree leaves, religious symbols, rugs, and handkerchiefs.

2. Previous Knowledge:

• Students have already encountered the concept of line symmetry in earlier classes.
• A figure has line symmetry if it can be folded along a line so that the two halves coincide perfectly.

3. Activities to Reinforce Understanding:

• Observing Nature: Identify symmetry in natural objects such as flowers, leaves, and beehives.
• Architecture and Engineering: Notice how buildings and structures often incorporate symmetrical designs.
• Creative Exercises:
  • Create colourful ink-dot devils to explore symmetry.
  • Make symmetrical paper-cut designs.
  • Compose a picture album showcasing symmetry found in different places.

4. Identifying Lines of Symmetry:

• Encourage students to enjoy finding lines (axes) of symmetry in various designs they come across.
• Example figures are provided where lines of symmetry are marked with dotted lines to help students visualize the concept.

5. Strengthening Symmetry Concepts:

• Further studies and examples are provided to deepen the understanding of symmetry.
• Figures: Study figures (Fig 12.1 i to iv) that show symmetry lines.

Explanation of Key Concepts in 12.1

1. Line Symmetry:

• A figure has line symmetry if there is a line (axis) where the figure can be folded, resulting in two coinciding parts.

2. Symmetry in Nature and Human-Made Objects:

• Nature: Bees use hexagonal symmetry in their hives, flowers often have radial symmetry, and leaves can have bilateral symmetry.
• Human-Made: Architecture often uses symmetry for aesthetic and structural purposes, designers use symmetry in patterns, and engineers may use it in creating balanced structures.

3. Interactive Learning:

• Ink-Dot Devils: Dropping ink on paper and folding it creates symmetrical patterns, helping visualize how symmetry works.
• Paper-Cut Designs: Cutting paper into symmetrical designs by folding it demonstrates how symmetry is created.
• Picture Album: Collecting images that show symmetry helps reinforce the concept through real-world examples.

4. Visual Learning with Figures:

• Figures in the textbook show examples of symmetrical shapes with their lines of symmetry marked, aiding in visual learning and understanding.

5. Concept of Mirror Reflection:

• Symmetry is closely related to mirror reflections, where one half of a shape is the mirror image of the other half.
• A mirror line helps visualize a line of symmetry.

Examples and Exercises:

• Figure Examples:

  • Fig 12.1 (i) to (iv): Illustrations showing different shapes with lines of symmetry marked.
  • Fig 12.2: An equilateral triangle with equal sides and angles, showing it’s a regular polygon.
  • Fig 12.3: A square with equal sides and right angles, showing it’s a regular polygon.
  • Fig 12.4 and Fig 12.5: Regular pentagon and hexagon with equal sides and specific angle measures.
  • Fig 12.6 (i) – (iv): Various regular polygons with multiple lines of symmetry.

• Mirror Reflection Examples:

  • Fig 12.7 and Fig 12.8: Show how one half of a shape can be a mirror image of the other, emphasizing the concept of symmetry.
  • Fig 12.9: Illustrates the left-right orientation change in mirror reflection.

• Interactive Example:

  • Fig 12.10: A punching game where a folded sheet with punched holes demonstrates lines of symmetry.

12.2 Lines of Symmetry for Regular Polygons

In this section, we delve deeper into understanding the lines of symmetry specifically for regular polygons. A regular polygon is defined as a polygon with all sides and angles equal. Here’s a detailed breakdown:

1. Definition of a Polygon:

• A polygon is a closed figure made up of several line segments.
• The simplest polygon, with the fewest line segments, is a triangle.

2. Definition of a Regular Polygon:

• A polygon is regular if all its sides are of equal length and all its angles are of equal measure.
• Examples include the equilateral triangle and the square.

3. Examples of Regular Polygons:

• Equilateral Triangle:

  • Each side has the same length.
  • Each angle measures 60 degrees.
  • It is a regular polygon with three sides. (See Fig 12.2)

• Square:

  • All sides are of equal length.
  • Each angle is a right angle (90 degrees).
  • Its diagonals are perpendicular bisectors of each other. (See Fig 12.3)

• Regular Pentagon:

  • All sides are equal in length.
  • Each angle measures 108 degrees.
  • It is a regular polygon with five sides. (See Fig 12.4)

• Regular Hexagon:

  • All sides are equal in length.
  • Each angle measures 120 degrees.
  • It is a regular polygon with six sides. (See Fig 12.5)

4. Lines of Symmetry in Regular Polygons:

• A regular polygon has as many lines of symmetry as it has sides.
• For example, an equilateral triangle has three lines of symmetry, a square has four, a regular pentagon has five, and a regular hexagon has six. (See Fig 12.6 i – iv)

5. Symmetry and Mirror Reflection:

• The concept of line symmetry is closely related to mirror reflection.
• A shape has line symmetry if one half of it is the mirror image of the other half.
• A mirror line helps to visualize the line of symmetry. (See Fig 12.7 and Fig 12.8)

6. Mirror Line Examples:

• Incorrect Mirror Line: If a line doesn’t divide a shape into two mirror-image halves, it is not a line of symmetry. (See the left part of Fig 12.8)
• Correct Mirror Line: A line that perfectly divides a shape into two mirror-image halves is a line of symmetry. (See the right part of Fig 12.8)

7. Interactive Learning:

• Punching Game:
  • Fold a sheet into two halves.
  • Punch holes through the symmetric fold.
  • When the paper is unfolded, the holes reflect the line of symmetry. (See Fig 12.10)
  • This activity helps visualize lines of symmetry and understand their properties.

8. Left-Right Orientation in Mirror Reflection:

• Mirror reflections can change the left-right orientation of objects.
• The shape remains the same, but its orientation is reversed. (See Fig 12.9)

Summary:

• Regular polygons are symmetrical figures with equal sides and angles.
• Each regular polygon has multiple lines of symmetry, equal to the number of its sides.
• The concept of line symmetry is linked to mirror reflections, where one half of a shape is the mirror image of the other half.
• Activities like paper folding and punching holes can help visualize and understand lines of symmetry.

12.3 Rotational Symmetry

In this section, we explore the concept of rotational symmetry, an important type of symmetry that is different from line symmetry. Here’s a detailed breakdown:

1. Definition of Rotational Symmetry:

• A figure has rotational symmetry if it can be rotated (less than a full circle of 360 degrees) around a central point and still look exactly the same.
• The central point around which the figure is rotated is called the center of rotation.

2. Order of Rotational Symmetry:

• The number of times a figure fits onto itself during a complete rotation of 360 degrees is called the order of rotational symmetry.
• For example, if a figure looks the same 3 times during a 360-degree rotation, its order of rotational symmetry is 3.

3. Examples of Rotational Symmetry:

• Equilateral Triangle:
  • An equilateral triangle has rotational symmetry of order 3.
  • It fits onto itself 3 times during a 360-degree rotation. (120 degrees each time)
• Square:
  • A square has rotational symmetry of order 4.
  • It fits onto itself 4 times during a 360-degree rotation. (90 degrees each time)
• Regular Pentagon:
  • A regular pentagon has rotational symmetry of order 5.
  • It fits onto itself 5 times during a 360-degree rotation. (72 degrees each time)
• Regular Hexagon:
  • A regular hexagon has rotational symmetry of order 6.
  • It fits onto itself 6 times during a 360-degree rotation. (60 degrees each time)

4. Identifying Rotational Symmetry:

• To determine if a figure has rotational symmetry and its order:
  • Identify the center of rotation.
  • Rotate the figure around this center and see how many times it fits onto itself within a full 360-degree turn.

5. Degrees of Rotation:

• The smallest angle at which the figure looks the same is called the angle of rotation.
• For a figure with rotational symmetry of order $n$, the angle of rotation is 360∘nfrac{360^circ}{n}n360∘​.
• For example:
  • For an equilateral triangle (order 3), the angle of rotation is 360∘3=120∘frac{360^circ}{3} = 120^circ3360∘​=120∘.
  • For a square (order 4), the angle of rotation is 360∘4=90∘frac{360^circ}{4} = 90^circ4360∘​=90∘.

6. Symmetry in Nature and Objects:

• Rotational symmetry is commonly found in nature and in various man-made objects:
  • Flowers: Many flowers exhibit rotational symmetry, like the daisy with petals arranged around the center.
  • Wheels: Car wheels, bicycle wheels, and other circular objects typically have rotational symmetry
  • .

7. Activity: Investigating Rotational Symmetry:

• Paper Folding and Rotation:
  • Draw a regular polygon (like a hexagon) on paper.
  • Cut it out and find its center.
  • Rotate the polygon around its center and observe how many times it fits onto itself within a 360-degree turn.