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Cbse Class 6 Maths Notes Chapter 9 Symmetry
Symmetry is an interesting concept that plays a crucial role in CBSE Class 6 Maths, art, nature, and even in our daily lives. In simple terms, symmetry refers to the balanced and proportional arrangement of parts on either side of a central point or axis. It helps us understand how shapes and objects can be mirrored, rotated, or reflected in different ways. In this blog, we’ll explore the different types of symmetry, including Line of Symmetry, Reflection, Rotational Symmetry, and the Symmetries of a Circle.
1.0Symmetry Class 6 Maths Chapter 9 - Revision Notes
What is Symmetry?
Symmetry can be described as the characteristic of an object or shape that allows it to be divided into two or more identical parts. These parts can either be mirrored or rotated around a fixed point or axis. Symmetry can be observed in various natural and man-made objects, from the wings of butterflies to the architectural designs of buildings.
Line of Symmetry
The Line of Symmetry is an imaginary line that splits a shape into two identical halves. If you fold a shape along this line, the two halves will overlap perfectly. This line can be vertical, horizontal, or diagonal, depending on the shape.
- For example, a square has four lines of symmetry (two diagonals and two perpendicular bisectors), whereas a circle has infinite lines of symmetry because it can be folded along any line passing through its centre.
Example:
Reflection Symmetry
Reflection symmetry, also known as mirror symmetry, occurs when one half of an object is a mirror image of the other half. This is possible when there is a line of symmetry dividing the object into two identical parts.
- Real-life example: The human face exhibits reflection symmetry when divided down the middle.
- A butterfly’s wings also show reflection symmetry, where each wing is a mirror image of the other.
Reflection symmetry is common in many natural forms and shapes, where the two halves mirror each other across a central line.
Rotational Symmetry
A shape has rotational symmetry if it can be rotated (less than a full circle) around a central point and still look exactly the same. The number of times a shape can fit into its original position during a complete rotation is called its order of rotational symmetry.
- For example, a regular hexagon has 6 lines of rotational symmetry, because it can be rotated by 60°, 120°, 180°, 240°, 300°, and 360°, and it will still look the same at each of these positions.
- A regular square has 4 lines of rotational symmetry, as it matches its original shape at 90°, 180°, 270°, and 360° rotations.
Symmetries of a Circle
- A circle is a special shape that exhibits infinite lines of symmetry. This means that you can draw a line through the centre of the circle in any direction, and the two halves will always be identical. The symmetry of a circle is a perfect example of how the concept of symmetry extends to continuous shapes.
- A circle also exhibits rotational symmetry of infinite order because it can be rotated by any angle, and it will always appear the same. Whether you rotate a circle by 1°, 90°, or 180°, the circle remains unchanged.
- This infinite symmetry makes the circle a unique and fundamental shape in geometry and real-world design.
Applications of Symmetry
Symmetry isn’t just an abstract concept; it has real-world applications in various fields:
- Art and Architecture: Symmetry plays a vital role in designing visually pleasing structures, from classical buildings to modern skyscrapers. The Parthenon in Greece is a perfect example of symmetry in architecture.
- Nature: The symmetry found in flowers, leaves, animal bodies, and even galaxies showcases the beauty of natural designs.
- Engineering: Symmetry helps in designing efficient mechanical parts and structures, such as gears and wheels, ensuring balance and strength.
Key Concepts to Remember
- Symmetry: A figure has symmetry when its parts follow a definite pattern. For example, a butterfly has two wings that look like mirror images of each other.
- Line of symmetry: A line of symmetry divides a figure into two parts that overlap exactly when folded along that line. Both parts must match perfectly.
- Mirror halves: The two parts formed by a line of symmetry are called mirror halves. If the parts do not overlap completely when folded, the line is not a line of symmetry.
- Multiple lines of symmetry: Some figures have more than one line of symmetry. A square has four lines of symmetry, while an equilateral triangle has three.
- Symmetry in a rectangle: A rectangle that is not a square has two lines of symmetry: one horizontal and one vertical through its centre. Its diagonals are not lines of symmetry.
- Reflection symmetry: A figure has reflection symmetry if one part reflects onto the other across a line of symmetry. The reflected parts match in shape and size.
- Making symmetrical shapes: Folding paper, cutting shapes and making ink patterns are simple ways to create symmetrical figures. The fold often acts as the line of symmetry.
- Completing a symmetrical figure: When a line of symmetry is given, the missing part must be drawn as a mirror image of the existing part. Each point should be the same distance from the line as its matching point.
- Rotational symmetry: A figure has rotational symmetry if it looks exactly the same after a turn of less than 360° around a fixed point.
- Centre of rotation: The fixed point around which a figure turns is called its centre of rotation.
- Angle of rotational symmetry: The angle through which a figure turns and matches its original position is called an angle of rotational symmetry. For example, a square matches itself after a 90° turn.
- Angles of symmetry of a square: A square matches its original position after turns of 90°, 180°, 270° and 360°. The turns below 360° show its rotational symmetry.
- Rotational symmetry of radial patterns: Radial patterns have parts arranged around a central point. For the pattern to match after a turn, its arms must be arranged at equal angles.
- Rotational symmetry of a circle: A circle looks the same when rotated through any angle about its centre. Every diameter is also a line of symmetry.
- Figures with different types of symmetry: A figure may have reflection symmetry, rotational symmetry, both or neither. A regular shape should be checked rather than judged only by its appearance.
- Symmetry in everyday objects: Flowers, rangoli designs, wheels, pinwheels, buildings and decorative patterns help us recognise symmetry in our surroundings.
2.0Important Formulas – Symmetry
This chapter is mainly about recognising and drawing symmetrical figures. These properties and rules will help you solve questions from the chapter.
Finding Lines of Symmetry
To find the lines of symmetry of a shape, imagine folding it along a possible line. If both halves overlap exactly, that line is a line of symmetry.
Examples:
- A square has 4 lines of symmetry.
- A rectangle that is not a square has 2 lines of symmetry.
- An equilateral triangle has 3 lines of symmetry.
- A circle has infinitely many lines of symmetry because every diameter divides it into matching halves.
Understanding Reflection Symmetry
Reflection symmetry works like looking into a mirror. The reflected part must match the original part.
To complete a figure when its line of symmetry is given:
- Identify the line of symmetry.
- Look at the position of each point on the existing side.
- Mark its matching point on the other side, at the same perpendicular distance from the line.
- Join the points in the same order to complete the figure.
For example, if a point lies two squares to the left of a vertical symmetry line, its matching point must lie two squares to the right.
Finding Angles of Rotational Symmetry
To check rotational symmetry, turn the figure around its centre and observe when it matches its original position.
For a square, the angles of symmetry are:
- 90° – quarter turn
- 180° – half turn
- 270° – three-quarter turn
- 360° – complete turn
The smallest angle of rotational symmetry is 90°.
Remember that a complete rotation of 360° always returns a figure to its original position. To find rotational symmetry , see if the shape is the same when rotated greater than 0° and less than 360°.
Finding Symmetry in Radial Patterns
Radial patterns have arms or parts arranged around a central point. When the arms are equally spaced, the smallest angle of rotation can be found by dividing 360° by the number of arms.
Examples:
- 2 equally spaced arms: 360° ÷ 2 = 180°
- 3 equally spaced arms: 360° ÷ 3 = 120°
- 4 equally spaced arms: 360° ÷ 4 = 90°
- 5 equally spaced arms: 360° ÷ 5 = 72°
This rule applies when the arms are identical and arranged at equal angles around the centre.
3.0Learning Guide for Class 6 Maths Symmetry – Chapter 9
- Checking symmetry by folding: Draw a few shapes on paper and fold each one along a possible line of symmetry. Check whether the two halves overlap completely. Try more than one fold for shapes such as squares.
- Finding lines of symmetry in shapes: Draw a square, rectangle, equilateral triangle and circle. Mark all their lines of symmetry and compare the number of lines in each shape.
- Understanding mirror images: Draw a vertical line on squared paper and make a simple shape on one side. Complete its mirror image on the other side by counting the same number of squares from the line.
- Making ink patterns: Fold a sheet of paper in half, place a few drops of paint on one side and press the halves together. Open the paper and identify the line of symmetry in the pattern.
- Trying paper cutting: Fold a sheet of paper and cut a small shape along the folded edges. Open the paper and observe the complete design. Identify the line along which the paper was folded.
- Playing the punching game: Fold a sheet of paper and punch holes in it. Unfold the paper and observe where the holes appear. Use the matching positions to identify the fold line.
- Completing shapes on a grid: Copy a half-completed figure onto squared paper. If a line of symmetry is shown, draw the missing half by placing each point at the same distance from the line.
- Comparing a square and a rectangle: Draw both shapes and mark their lines of symmetry. Check why a square has four lines while a rectangle that is not a square has only two.
- Exploring rotational symmetry: Cut out a square and rotate it around its centre. Check when it matches its original position. Repeat with other shapes and record the angles at which they match.
- Making radial patterns: Draw three or four identical arms around a central point. Keep the angles between neighbouring arms equal and rotate the figure to check whether it matches itself.
- Finding symmetry in a circle: Draw a circle and several diameters. Check how each diameter divides the circle into two matching halves. Then rotate the circle and observe that it looks unchanged.
- Looking for symmetry around you: Observe flowers, rangoli designs, wheels, fans and buildings. Identify whether each example has reflection symmetry, rotational symmetry or both.
- Creating symmetrical tile patterns: Use identical paper shapes or tiles to make a repeating design. Try arranging them so the finished pattern has one or more lines of symmetry.
- Checking your answers: Do not assume that a line is a line of symmetry just because it passes through the centre. Check whether both halves match exactly. For rotational symmetry, check whether the whole figure matches after the turn.
4.0Explore Chapter-wise Class 6 Maths Notes
5.0Additional Study Materials for Class 6 Maths
6.0Find the Right Study Resources for Class 6
7.0Benefits of CBSE Notes for Class 6 Maths Chapter 9 - Symmetry
- Practice Questions and Solutions: Many CBSE Notes include practice questions with solutions, allowing you to test your understanding and identify areas where you need more practice.
- Time-Saving: Instead of going through the entire NCERT Solutions textbook, notes provide a summarised version of the chapter, saving you valuable study time.
- Improved Problem-Solving Skills: By understanding the core concepts through the notes, you can approach problem-solving in a more structured and confident way.
- Better Exam Preparation: Comprehensive CBSE Notes for Class 6 Maths ensure you are well-prepared for your exams by covering all the essential topics and types of questions.
- Building a Strong Foundation: A solid understanding of symmetry in CBSE Class 6 forms a crucial foundation for more advanced geometry concepts in higher classes.