Class 6 Maths – Chapter 9: Symmetry
Complete NCERT Solutions with All Exercises
Master Symmetry Concepts and Patterns
Learning Objectives - Chapter 9: Symmetry
- Understand line symmetry and reflection symmetry
- Identify symmetrical objects in daily life
- Find lines of symmetry in various shapes
- Understand rotational symmetry
- Complete symmetrical figures and patterns
- Create symmetrical designs and artwork
Important Concepts - Symmetry
Line Symmetry
Mirror reflection
Two identical halves
Fold along the line
Rotational Symmetry
Looks same after rotation
Order of rotation
Angle of rotation
Reflection Symmetry
Mirror image
Line of symmetry
Bilateral symmetry
Point Symmetry
Same from all directions
180° rotation
Radial symmetry
Key Symmetry Principles:
Line of Symmetry: Divides figure into mirror images
Rotational Symmetry: Figure fits into itself during rotation
Order of Rotation: Number of times figure matches itself in 360°
Angle of Rotation: 360° ÷ Order of rotation
Point Symmetry: Looks same when rotated 180°
Exploring Symmetry! ⇌
Welcome to Chapter 9: Symmetry! This chapter introduces you to the beautiful world of symmetry, which we see all around us in nature, art, architecture, and everyday objects.
We'll explore different types of symmetry - line symmetry (reflection symmetry), rotational symmetry, and point symmetry. You'll learn to identify lines of symmetry in various shapes, complete symmetrical figures, and understand how symmetry makes objects balanced and aesthetically pleasing.
By mastering symmetry, you'll develop visual thinking skills and a deeper appreciation for patterns in mathematics, art, and the natural world!
A
Vertical symmetry
A
Mirror image
Line Symmetry
Mirror reflection across a line
Rotational Symmetry
Looks same after rotation
Reflection Symmetry
Bilateral symmetry
Point Symmetry
Same from all directions
NCERT Textbook Exercises & Solutions Class 6 Chapter 9: Symmetry
Complete step-by-step solutions for all NCERT textbook exercises
1. Do you see any line of symmetry in the figures at the start of the chapter? What about in the picture of the cloud?
💡 Hint: A line of symmetry divides a figure into two identical mirror halves.
Step-by-step Explanation:
Step 1: Examine each figure
Look at the flower, rangoli, butterfly, pinwheel, and cloud figures.
Look at the flower, rangoli, butterfly, pinwheel, and cloud figures.
Step 2: Check for symmetry
Flower: Has multiple lines of symmetry (6)
Rangoli: Has multiple lines of symmetry (4)
Butterfly: Has one line of symmetry down the center
Pinwheel: No line symmetry (has rotational symmetry)
Cloud: No line symmetry (irregular shape)
Flower: Has multiple lines of symmetry (6)
Rangoli: Has multiple lines of symmetry (4)
Butterfly: Has one line of symmetry down the center
Pinwheel: No line symmetry (has rotational symmetry)
Cloud: No line symmetry (irregular shape)
Answer: Yes, the flower has 6 lines, rangoli has 4 lines, butterfly has 1 line of symmetry. The pinwheel and cloud have no lines of symmetry.
2. For each of the following figures, identify the line(s) of symmetry if it exists.
💡 Hint: Draw imaginary lines through each figure to see if both halves match exactly.
Step-by-step Explanation:
Step 1: Understand line symmetry
A line of symmetry divides a figure into two identical mirror images.
A line of symmetry divides a figure into two identical mirror images.
Step 2: Examine each figure
Check for vertical, horizontal, and diagonal lines that could be lines of symmetry.
Check for vertical, horizontal, and diagonal lines that could be lines of symmetry.
Step 3: Draw the lines
For each figure, draw the lines where the figure can be folded to match exactly.
For each figure, draw the lines where the figure can be folded to match exactly.
Answer: The lines of symmetry vary for each figure as per their shapes.
3. Is there any other way to fold the square so that the two halves overlap? How many lines of symmetry does the square shape have?
💡 Hint: A square has multiple lines of symmetry through different axes.
Step-by-step Explanation:
Step 1: Identify possible fold lines
A square can be folded along vertical, horizontal, and both diagonal axes.
A square can be folded along vertical, horizontal, and both diagonal axes.
Step 2: Count the lines of symmetry
Vertical line: 1
Horizontal line: 1
Diagonals: 2
Total: 4 lines of symmetry
Vertical line: 1
Horizontal line: 1
Diagonals: 2
Total: 4 lines of symmetry
Answer: No other ways to fold. A square has 4 lines of symmetry.
4. We saw that the diagonal of a square is also a line of symmetry. Let us take a rectangle that is not a square. Is its diagonal a line of symmetry?
💡 Hint: Try folding a rectangle along its diagonal.
Step-by-step Explanation:
Step 1: Understand the difference
A square has all sides equal, while a rectangle has only opposite sides equal.
A square has all sides equal, while a rectangle has only opposite sides equal.
Step 2: Test diagonal symmetry
Fold a non-square rectangle along its diagonal. The two halves won't match exactly.
Fold a non-square rectangle along its diagonal. The two halves won't match exactly.
Step 3: Conclusion
The diagonal of a non-square rectangle is not a line of symmetry.
The diagonal of a non-square rectangle is not a line of symmetry.
Answer: No, the diagonal of a non-square rectangle is not a line of symmetry.
5. What if we reflect along the diagonal from A to C? Where do points A, B, C and D go? What if we reflect along the horizontal line of symmetry?
💡 Hint: In reflection, points swap positions across the line of symmetry.
Step-by-step Explanation:
Step 1: Diagonal reflection (A to C)
Points on the line (A and C) stay in place.
Point B and D swap positions.
Points on the line (A and C) stay in place.
Point B and D swap positions.
Step 2: Horizontal reflection
Points on the horizontal line stay in place.
Top and bottom points swap positions.
Points on the horizontal line stay in place.
Top and bottom points swap positions.
Answer: Diagonal reflection: A and C stay, B and D swap. Horizontal reflection: Points on horizontal line stay, others swap vertically.
6. In each of the following figures, a hole was punched in a folded square sheet of paper and then the paper was unfolded. Identify the line along which the paper was folded.
💡 Hint: The pattern of holes reveals the fold lines.
Step-by-step Explanation:
Step 1: Analyze the hole pattern
The holes will be symmetric with respect to the fold line.
The holes will be symmetric with respect to the fold line.
Step 2: Identify the symmetry
For a single fold, holes will appear in pairs symmetric about the fold line.
For a single fold, holes will appear in pairs symmetric about the fold line.
Step 3: Determine the fold line
Draw the line that would make the hole pattern symmetric.
Draw the line that would make the hole pattern symmetric.
Answer: The fold lines are the lines of symmetry for the hole patterns.
7. Given the line(s) of symmetry, find the other hole(s).
💡 Hint: Reflect the given holes across the symmetry lines.
Step-by-step Explanation:
Step 1: Understand reflection
Each hole has a mirror image across each line of symmetry.
Each hole has a mirror image across each line of symmetry.
Step 2: Locate mirror images
For each given hole, find its reflection across each symmetry line.
For each given hole, find its reflection across each symmetry line.
Step 3: Mark the holes
Mark all the reflected positions to complete the pattern.
Mark all the reflected positions to complete the pattern.
Answer: The other holes are the mirror images of the given holes across the symmetry lines.
8. After each of the following cuts, predict the shape of the hole when the paper is opened.
💡 Hint: Consider how folding affects the final cut shape.
Step-by-step Explanation:
Step 1: Analyze the fold pattern
Determine how many times and in what directions the paper is folded.
Determine how many times and in what directions the paper is folded.
Step 2: Predict the cut pattern
The final shape will have symmetry corresponding to the folds.
The final shape will have symmetry corresponding to the folds.
Step 3: Visualize the result
Imagine unfolding the paper to see the complete pattern.
Imagine unfolding the paper to see the complete pattern.
Answer: The final hole shapes will have symmetry based on the folding pattern.
9. How many lines of symmetry do these shapes have? a. A shape with 4 lines of symmetry and 8 lines of symmetry. b. A triangle with equal sides and equal angles. c. A hexagon with equal sides and equal angles.
💡 Hint: Regular polygons have multiple lines of symmetry.
Step-by-step Explanation:
Step 1a: Shape with 4 and 8 lines
Some shapes can have both 4 and 8 lines of symmetry if they have multiple symmetric elements.
Some shapes can have both 4 and 8 lines of symmetry if they have multiple symmetric elements.
Step 1b: Equilateral triangle
An equilateral triangle has 3 lines of symmetry (through each vertex to midpoint of opposite side).
An equilateral triangle has 3 lines of symmetry (through each vertex to midpoint of opposite side).
Step 1c: Regular hexagon
A regular hexagon has 6 lines of symmetry (through opposite vertices and through midpoints of opposite sides).
A regular hexagon has 6 lines of symmetry (through opposite vertices and through midpoints of opposite sides).
Answer: a. 4 and 8 lines, b. 3 lines, c. 6 lines of symmetry.
10. Trace each figure and draw the lines of symmetry, if any.
💡 Hint: Look for lines that divide the figure into mirror images.
Step-by-step Explanation:
Step 1: Examine each figure carefully
Look for any lines that would create identical halves.
Look for any lines that would create identical halves.
Step 2: Test potential symmetry lines
Draw imaginary lines and check if both sides match.
Draw imaginary lines and check if both sides match.
Step 3: Draw the confirmed lines
Mark all valid lines of symmetry on each figure.
Mark all valid lines of symmetry on each figure.
Answer: Draw the appropriate lines of symmetry for each figure.
11. Find the lines of symmetry for the kolam below.
💡 Hint: Kolam patterns often have multiple symmetric elements.
Step-by-step Explanation:
Step 1: Analyze the kolam pattern
Look for repeating elements and symmetric arrangements.
Look for repeating elements and symmetric arrangements.
Step 2: Identify symmetry lines
Common lines include vertical, horizontal, and diagonal axes.
Common lines include vertical, horizontal, and diagonal axes.
Step 3: Verify each line
Ensure each proposed line truly creates mirror images.
Ensure each proposed line truly creates mirror images.
Answer: The kolam has multiple lines of symmetry through its center.
12. Draw: a. A triangle with exactly one line of symmetry. b. A triangle with exactly three lines of symmetry. c. A triangle with no line of symmetry. Is it possible to draw a triangle with exactly two lines of symmetry?
💡 Hint: Consider different types of triangles - isosceles, equilateral, scalene.
Step-by-step Explanation:
Step 1a: One line of symmetry
Draw an isosceles triangle (only two equal sides).
Draw an isosceles triangle (only two equal sides).
Step 1b: Three lines of symmetry
Draw an equilateral triangle (all sides equal).
Draw an equilateral triangle (all sides equal).
Step 1c: No line of symmetry
Draw a scalene triangle (all sides different).
Draw a scalene triangle (all sides different).
Step 2: Two lines of symmetry?
No, a triangle cannot have exactly two lines of symmetry. If it has two, it must have three (equilateral).
No, a triangle cannot have exactly two lines of symmetry. If it has two, it must have three (equilateral).
Answer: a. Isosceles triangle, b. Equilateral triangle, c. Scalene triangle. No, a triangle cannot have exactly two lines of symmetry.
13. Draw: a. A figure with exactly one line of symmetry. b. A figure with exactly two lines of symmetry. c. A figure with exactly four lines of symmetry. In each case, the figure should contain at least one curved boundary.
💡 Hint: Think of shapes like hearts, ovals, and special curved figures.
Step-by-step Explanation:
Step 1a: One line of symmetry
Draw a heart shape or a teardrop shape.
Draw a heart shape or a teardrop shape.
Step 1b: Two lines of symmetry
Draw an oval or ellipse.
Draw an oval or ellipse.
Step 1c: Four lines of symmetry
Draw a curved pattern with four-fold symmetry, like a clover shape.
Draw a curved pattern with four-fold symmetry, like a clover shape.
Answer: Draw appropriate curved figures with the specified number of symmetry lines.
14. Copy the following on squared paper. Complete them so that the blue line is a line of symmetry.
💡 Hint: Reflect the existing parts across the symmetry line.
Step-by-step Explanation:
Step 1: Understand reflection
Each point on one side has a mirror image on the other side.
Each point on one side has a mirror image on the other side.
Step 2: Locate key points
Identify important points on the given part of the figure.
Identify important points on the given part of the figure.
Step 3: Reflect across the line
Find the mirror images of these points and connect them appropriately.
Find the mirror images of these points and connect them appropriately.
Answer: Complete each figure by reflecting the given part across the blue line.
15. Copy the following drawing on squared paper. Complete each one of them so that the resulting figure has the two blue lines as lines of symmetry.
💡 Hint: Reflect across both symmetry lines to complete the figure.
Step-by-step Explanation:
Step 1: Understand double symmetry
The figure must be symmetric about both given lines.
The figure must be symmetric about both given lines.
Step 2: Reflect across first line
Create the mirror image across the first blue line.
Create the mirror image across the first blue line.
Step 3: Reflect across second line
Ensure the resulting figure is also symmetric about the second blue line.
Ensure the resulting figure is also symmetric about the second blue line.
Answer: Complete each figure to be symmetric about both blue lines.
16. Copy the following on a dot grid. For each figure draw two more lines to make a shape that has a line of symmetry.
💡 Hint: Add lines to create a symmetric pattern.
Step-by-step Explanation:
Step 1: Analyze the given figure
Understand what parts are already present.
Understand what parts are already present.
Step 2: Plan the symmetric shape
Decide what symmetric shape you can create by adding two lines.
Decide what symmetric shape you can create by adding two lines.
Step 3: Draw the additional lines
Add two lines that complete a symmetric figure.
Add two lines that complete a symmetric figure.
Answer: Add two lines to each figure to create a symmetric shape.
17. Can you draw a figure with radial arms that has a) exactly 5 angles of symmetry, b) 6 angles of symmetry? Also find the angles of symmetry in each case.
💡 Hint: Divide 360° by the number of symmetric positions.
Step-by-step Explanation:
Step 1a: 5 angles of symmetry
Smallest angle = 360° ÷ 5 = 72°
Angles: 72°, 144°, 216°, 288°, 360°
Smallest angle = 360° ÷ 5 = 72°
Angles: 72°, 144°, 216°, 288°, 360°
Step 1b: 6 angles of symmetry
Smallest angle = 360° ÷ 6 = 60°
Angles: 60°, 120°, 180°, 240°, 300°, 360°
Smallest angle = 360° ÷ 6 = 60°
Angles: 60°, 120°, 180°, 240°, 300°, 360°
Answer: a. Angles: 72°, 144°, 216°, 288°, 360°; b. Angles: 60°, 120°, 180°, 240°, 300°, 360°
18. Consider a figure with radial arms having exactly 7 angles of symmetry. What will be its smallest angle of symmetry? Is the number of degrees a whole number in this case? If not, express it as a mixed fraction.
💡 Hint: Divide 360 by 7.
Step-by-step Explanation:
Step 1: Calculate smallest angle
360° ÷ 7 = 51³⁄₇°
360° ÷ 7 = 51³⁄₇°
Step 2: Check if whole number
No, 360 is not divisible by 7, so it's not a whole number.
No, 360 is not divisible by 7, so it's not a whole number.
Step 3: Express as mixed fraction
51³⁄₇° (fifty-one and three-sevenths degrees)
51³⁄₇° (fifty-one and three-sevenths degrees)
Answer: Smallest angle = 51³⁄₇°. No, it's not a whole number.
19. Find the angles of symmetry for the given figures about the point marked.
💡 Hint: Look for the smallest rotation that makes the figure look the same.
Step-by-step Explanation:
Step 1a: Figure with 4-fold symmetry
Angles: 90°, 180°, 270°, 360°
Angles: 90°, 180°, 270°, 360°
Step 1b: Figure with no rotational symmetry
Only 360° (full rotation)
Only 360° (full rotation)
Step 1c: Figure with 2-fold symmetry
Angles: 180°, 360°
Angles: 180°, 360°
Answer: a. 90°, 180°, 270°, 360°; b. 360°; c. 180°, 360°
20. Which of the following figures have more than one angle of symmetry?
💡 Hint: Figures with rotational symmetry have multiple angles of symmetry.
Step-by-step Explanation:
Step 1: Understand angles of symmetry
These are angles less than 360° for which the figure looks the same after rotation.
These are angles less than 360° for which the figure looks the same after rotation.
Step 2: Check each figure
Identify figures that have rotational symmetry of order 2 or more.
Identify figures that have rotational symmetry of order 2 or more.
Step 3: List qualifying figures
Figures with order 2, 3, 4, etc. have multiple angles of symmetry.
Figures with order 2, 3, 4, etc. have multiple angles of symmetry.
Answer: Figures with rotational symmetry of order 2 or more have multiple angles of symmetry.
21. Give the order of rotational symmetry for each figure.
💡 Hint: Order of symmetry is how many times the figure matches itself in a full rotation.
Step-by-step Explanation:
Step 1: Understand order of symmetry
Order = 360° ÷ smallest angle of symmetry
Order = 360° ÷ smallest angle of symmetry
Step 2: Calculate for each figure
Figure with 180° symmetry: order = 360÷180 = 2
Figure with 90° symmetry: order = 360÷90 = 4
Figure with 60° symmetry: order = 360÷60 = 6
Figure with 120° symmetry: order = 360÷120 = 3
Figure with 72° symmetry: order = 360÷72 = 5
Figure with 180° symmetry: order = 360÷180 = 2
Figure with 90° symmetry: order = 360÷90 = 4
Figure with 60° symmetry: order = 360÷60 = 6
Figure with 120° symmetry: order = 360÷120 = 3
Figure with 72° symmetry: order = 360÷72 = 5
Answer: Orders: 2, 4, 6, 3, 4, 5
22. Color the sectors of the circle below so that the figure has i) 3 angles of symmetry, ii) 4 angles of symmetry, iii) what are the possible numbers of angles of symmetry you can obtain by coloring the sectors in different ways?
💡 Hint: Use repeating color patterns to create symmetry.
Step-by-step Explanation:
Step 1i: 3 angles of symmetry
Divide circle into 3 equal sectors and color them in a repeating pattern.
Divide circle into 3 equal sectors and color them in a repeating pattern.
Step 1ii: 4 angles of symmetry
Divide circle into 4 equal sectors and color them in a repeating pattern.
Divide circle into 4 equal sectors and color them in a repeating pattern.
Step 1iii: Possible numbers
Any divisor of 360: 1, 2, 3, 4, 5, 6, 8, 9, 10, 12, 15, 18, 20, 24, 30, 36, 40, 45, 60, 72, 90, 120, 180, 360
Any divisor of 360: 1, 2, 3, 4, 5, 6, 8, 9, 10, 12, 15, 18, 20, 24, 30, 36, 40, 45, 60, 72, 90, 120, 180, 360
Answer: i) Use 3-fold pattern, ii) Use 4-fold pattern, iii) All divisors of 360 are possible.
23. Draw two figures other than a circle and a square that have both reflection symmetry and rotational symmetry.
💡 Hint: Think of regular polygons and symmetric patterns.
Step-by-step Explanation:
Step 1: Understand requirements
Need figures with both line symmetry and rotational symmetry.
Need figures with both line symmetry and rotational symmetry.
Step 2: Choose appropriate figures
Regular pentagon (5 lines of symmetry, order 5 rotational)
Regular hexagon (6 lines of symmetry, order 6 rotational)
Regular pentagon (5 lines of symmetry, order 5 rotational)
Regular hexagon (6 lines of symmetry, order 6 rotational)
Answer: Regular pentagon and regular hexagon have both types of symmetry.
24. Draw, wherever possible, a rough sketch of: a. A triangle with at least two lines of symmetry and at least two angles of symmetry. b. A triangle with only one line of symmetry but not having rotational symmetry. c. A quadrilateral with rotational symmetry but no reflection symmetry. d. A quadrilateral with reflection symmetry but not having rotational symmetry.
💡 Hint: Consider different types of triangles and quadrilaterals.
Step-by-step Explanation:
Step 1a: Triangle with ≥2 lines and angles of symmetry
Only equilateral triangle has 3 lines and order 3 rotational symmetry.
Only equilateral triangle has 3 lines and order 3 rotational symmetry.
Step 1b: Triangle with one line but no rotational symmetry
Isosceles triangle (has one line of symmetry but no rotational symmetry).
Isosceles triangle (has one line of symmetry but no rotational symmetry).
Step 1c: Quadrilateral with rotational but no reflection symmetry
Parallelogram (has 180° rotational symmetry but no line symmetry unless it's a rectangle).
Parallelogram (has 180° rotational symmetry but no line symmetry unless it's a rectangle).
Step 1d: Quadrilateral with reflection but no rotational symmetry
Isosceles trapezoid (has one line of symmetry but no rotational symmetry).
Isosceles trapezoid (has one line of symmetry but no rotational symmetry).
Answer: a. Equilateral triangle, b. Isosceles triangle, c. Parallelogram, d. Isosceles trapezoid.
25. In a figure, 60° is the smallest angle of symmetry. What are the other angles of symmetry of the figure?
💡 Hint: Angles of symmetry are multiples of the smallest angle.
Step-by-step Explanation:
Step 1: Understand the pattern
If smallest angle is A, then angles of symmetry are A, 2A, 3A, ... until 360°.
If smallest angle is A, then angles of symmetry are A, 2A, 3A, ... until 360°.
Step 2: Calculate for 60°
60°, 120°, 180°, 240°, 300°, 360°
60°, 120°, 180°, 240°, 300°, 360°
Answer: 120°, 180°, 240°, 300°, 360°
26. In the figure, 60° is an angle of symmetry. The figure has two angles of symmetry less than 60°. What is its smallest angle of symmetry?
💡 Hint: If 60° is the third smallest, what are the first two?
Step-by-step Explanation:
Step 1: Understand the sequence
If 60° is an angle of symmetry and there are two smaller ones, then the smallest must divide 60°.
If 60° is an angle of symmetry and there are two smaller ones, then the smallest must divide 60°.
Step 2: Find possible values
The two smaller angles could be 20° and 40° (since 3×20=60) or 30° and 60° (but 60° is given as third).
The two smaller angles could be 20° and 40° (since 3×20=60) or 30° and 60° (but 60° is given as third).
Step 3: Determine correct answer
If 60° is the third angle, then smallest is 20° (20°, 40°, 60°).
If 60° is the third angle, then smallest is 20° (20°, 40°, 60°).
Answer: Smallest angle of symmetry = 20°
27. Can we have a figure with rotational symmetry whose smallest angle of symmetry is a. 45°? b. 17°?
💡 Hint: 360° must be divisible by the smallest angle of symmetry.
Step-by-step Explanation:
Step 1: Check divisibility
For rotational symmetry, 360° must be divisible by the smallest angle.
For rotational symmetry, 360° must be divisible by the smallest angle.
Step 2a: 45°
360 ÷ 45 = 8 (integer) → Yes, possible
360 ÷ 45 = 8 (integer) → Yes, possible
Step 2b: 17°
360 ÷ 17 ≈ 21.176 (not integer) → No, not possible
360 ÷ 17 ≈ 21.176 (not integer) → No, not possible
Answer: a. Yes, b. No
28. This is a picture of the new Parliament Building in Delhi. a. Does the outer boundary of the picture have reflection symmetry? If so, draw the lines of symmetries. How many are they? b. Does it have rotational symmetry around its centre? If so, find the angles of rotational symmetry.
💡 Hint: The Parliament Building has triangular symmetry.
Step-by-step Explanation:
Step 1a: Reflection symmetry
Yes, the triangular shape has 3 lines of symmetry.
Yes, the triangular shape has 3 lines of symmetry.
Step 1b: Rotational symmetry
Yes, it has 3-fold rotational symmetry with angles 120°, 240°, 360°.
Yes, it has 3-fold rotational symmetry with angles 120°, 240°, 360°.
Answer: a. Yes, 3 lines of symmetry; b. Yes, angles: 120°, 240°, 360°
29. How many lines of symmetry do the shapes in the first shape sequence in Chapter 1, Table 3, the Regular Polygons, have? What number sequence do you get?
💡 Hint: Regular polygons have lines of symmetry equal to their number of sides.
Step-by-step Explanation:
Step 1: Regular polygons symmetry
A regular n-sided polygon has n lines of symmetry.
A regular n-sided polygon has n lines of symmetry.
Step 2: Create the sequence
Triangle: 3, Quadrilateral: 4, Pentagon: 5, Hexagon: 6, Heptagon: 7, Octagon: 8, Nonagon: 9, Decagon: 10
Triangle: 3, Quadrilateral: 4, Pentagon: 5, Hexagon: 6, Heptagon: 7, Octagon: 8, Nonagon: 9, Decagon: 10
Answer: 3, 4, 5, 6, 7, 8, 9, 10 - the counting number sequence starting from 3.
30. How many lines of symmetry do the shapes in the last shape sequence in Chapter 1, Table 3, the Koch Snowflake sequence, have? How many angles of symmetry?
💡 Hint: Koch snowflake has interesting symmetry properties.
Step-by-step Explanation:
Step 1: Koch snowflake symmetry
The Koch snowflake has 3 lines of symmetry and 3 angles of symmetry (120°, 240°, 360°).
The Koch snowflake has 3 lines of symmetry and 3 angles of symmetry (120°, 240°, 360°).
Step 2: Sequence pattern
This remains constant through the sequence: 3 lines and 3 angles of symmetry.
This remains constant through the sequence: 3 lines and 3 angles of symmetry.
Answer: Lines of symmetry: 3, 6, 6, 6, 6; Angles of symmetry: 3, 6, 6, 6, 6
31. How many lines of symmetry and angles of symmetry does Ashoka Chakra have?
💡 Hint: Count the spokes in the Ashoka Chakra.
Step-by-step Explanation:
Step 1: Analyze Ashoka Chakra
The Ashoka Chakra has 24 spokes arranged equally.
The Ashoka Chakra has 24 spokes arranged equally.
Step 2: Determine symmetry
It has 24 lines of symmetry (through each spoke and between spokes)
It has 24 angles of symmetry (360° ÷ 24 = 15° intervals)
It has 24 lines of symmetry (through each spoke and between spokes)
It has 24 angles of symmetry (360° ÷ 24 = 15° intervals)
Answer: 24 lines of symmetry and 24 angles of symmetry.
⇌ Historical Fact: Did you know that symmetry has been studied since ancient times? The ancient Greeks were fascinated by symmetry and used it extensively in their architecture, like the Parthenon. The concept of the "Golden Ratio" (approximately 1.618) was discovered by them and is found in many symmetrical patterns in nature!
🎯 Practical Activity: Create your own symmetrical art! Fold a paper in half, put drops of paint on one side, then fold and press. When you open it, you'll have a beautiful symmetrical pattern! You can also try cutting folded paper to create symmetrical shapes like snowflakes or butterflies.
20 Additional Practice Questions
Extra questions to master Symmetry
Multiple Choice Questions
- How many lines of symmetry does a square have?
A) 1 B) 2 C) 4 D) 8 - Which of the following has infinite lines of symmetry?
A) Square B) Rectangle C) Circle D) Triangle - What is the order of rotational symmetry of an equilateral triangle?
A) 1 B) 2 C) 3 D) 4 - Which letter of the English alphabet has both horizontal and vertical lines of symmetry?
A) A B) B C) H D) M - A figure has rotational symmetry of order 4. What is the angle of rotation?
A) 45° B) 60° C) 90° D) 120°
Fill in the Blanks
- A figure has _______ symmetry if it can be divided into two identical mirror halves.
- The number of times a figure fits into itself in one full rotation is called _______.
- A rectangle has _______ lines of symmetry.
- A regular pentagon has _______ lines of symmetry.
- A figure with rotational symmetry of order 2 looks the same after a rotation of _______.
True or False
- All triangles have at least one line of symmetry. (True/False)
- A square has more lines of symmetry than a rectangle. (True/False)
- Every figure with line symmetry also has rotational symmetry. (True/False)
- The letter 'S' has line symmetry. (True/False)
- A circle has infinite order of rotational symmetry. (True/False)
Short Answer Questions
- Draw a square and show all its lines of symmetry.
- Identify the number of lines of symmetry in a regular hexagon.
- Give two examples from nature that show symmetry.
- Draw the reflection of the word "MOM" across a vertical line.
- What is the difference between line symmetry and rotational symmetry?
Answer Key for Practice Questions
Multiple Choice:
- 1. C) 4
- 2. C) Circle
- 3. C) 3
- 4. C) H
- 5. C) 90°
Fill in the Blanks:
- 6. line
- 7. order of rotational symmetry
- 8. 2
- 9. 5
- 10. 180°
True or False:
- 11. False
- 12. True
- 13. False
- 14. False
- 15. True
Short Answers:
- 16. Square has 4 lines of symmetry (2 through midpoints, 2 through vertices)
- 17. Regular hexagon has 6 lines of symmetry
- 18. Butterfly, starfish, snowflake, human face, leaves (any two)
- 19. The reflection would look the same as "MOM" has vertical symmetry
- 20. Line symmetry: mirror reflection; Rotational symmetry: looks same after rotation
Chapter Summary - Symmetry
Key Concepts:
Symmetry: Symmetry is when one shape becomes exactly like another when you move it in some way - turn, flip, or slide.
Types of Symmetry:
- Line Symmetry (Reflection Symmetry): A figure has line symmetry if it can be divided into two identical mirror halves by a line. This line is called the line of symmetry.
- Rotational Symmetry: A figure has rotational symmetry if it looks the same after some rotation (less than 360°). The number of times it matches itself during a full rotation is called the order of rotational symmetry.
- Point Symmetry: A figure has point symmetry if it looks the same when rotated 180° around a central point.
Lines of Symmetry in Common Shapes:
- Square: 4 lines of symmetry
- Rectangle: 2 lines of symmetry
- Circle: Infinite lines of symmetry
- Equilateral Triangle: 3 lines of symmetry
- Isosceles Triangle: 1 line of symmetry
- Scalene Triangle: No lines of symmetry
- Regular Pentagon: 5 lines of symmetry
- Regular Hexagon: 6 lines of symmetry
Rotational Symmetry in Common Shapes:
- Square: Order 4 (90°, 180°, 270°, 360°)
- Rectangle: Order 2 (180°, 360°)
- Circle: Infinite order
- Equilateral Triangle: Order 3 (120°, 240°, 360°)
- Regular Pentagon: Order 5 (72°, 144°, 216°, 288°, 360°)
- Regular Hexagon: Order 6 (60°, 120°, 180°, 240°, 300°, 360°)
Symmetry in English Alphabet:
- Vertical Symmetry: A, H, I, M, O, T, U, V, W, X, Y
- Horizontal Symmetry: B, C, D, E, H, I, O, X
- Both Vertical and Horizontal: H, I, O, X
- No Symmetry: F, G, J, K, L, N, P, Q, R, S, Z
Applications:
Symmetry is used in:
- Art and design patterns
- Architecture and building design
- Nature (butterflies, flowers, snowflakes)
- Manufacturing and engineering
- Logo design and branding
- Textile and fashion design
- Photography composition
Teaching Guide & Notes
📚 Effective Teaching Strategies:
- Use actual mirrors to demonstrate line symmetry
- Organize symmetry hunts in the classroom or school
- Use paper folding and cutting activities
- Incorporate art projects with symmetrical designs
- Use pattern blocks and tangrams
🎯 Common Learning Gaps:
- Confusion between line symmetry and rotational symmetry
- Difficulty identifying all lines of symmetry in complex shapes
- Trouble understanding order of rotational symmetry
- Difficulty completing symmetrical figures
🔍 Key Concepts to Emphasize:
- The mirror test for line symmetry
- Difference between line and rotational symmetry
- Relationship between order of rotation and angle of rotation
- Symmetry in everyday objects and nature
- Practical applications of symmetry