NCERT Solutions for Class 6 Maths Chapter 13 — Symmetry

Key Concepts

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Exercise 13.1 — Line of Symmetry: Introduction and Identification

Question 1

Q: List any four symmetrical objects from your home or school.

Other examples include a pencil box, a pair of scissors, a geometry box, and a water bottle (when viewed from the front).

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Question 2

Q: For the given figure, identify the line(s) of symmetry.

Step 1: Try the vertical line through the centre. If the left half and the right half are identical, this is a line of symmetry. Step 2: Try the horizontal line through the centre. If the top half and bottom half match, this is another line of symmetry. Step 3: Try diagonal lines. Check if the figure coincides when folded along each diagonal.

A symmetrical figure may have one or more lines of symmetry depending on its shape. We count all lines along which the figure folds into matching halves.

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Question 3

Q: Identify the shapes given below and find the number of lines of symmetry for each.

(a) Equilateral triangle Solution: An equilateral triangle has all three sides equal and all three angles equal (60 degrees each).

• Fold along the line from one vertex to the midpoint of the opposite side: the two halves coincide.
• There are 3 such lines (one from each vertex to the midpoint of the opposite side).
• Number of lines of symmetry = 3

(b) Square Solution: A square has all four sides equal and all angles 90 degrees.

• Fold along the vertical line through the centre: halves match.
• Fold along the horizontal line through the centre: halves match.
• Fold along each diagonal: halves match.
• Number of lines of symmetry = 4

(c) Rectangle Solution: A rectangle has opposite sides equal and all angles 90 degrees.

• Fold along the vertical line through the centre: halves match.
• Fold along the horizontal line through the centre: halves match.
• Fold along the diagonals: halves do NOT match (diagonals are not lines of symmetry for a rectangle).
• Number of lines of symmetry = 2

(d) Circle Solution: A circle has every diameter as a line of symmetry.

• Fold along any diameter: the two semicircles match.
• Since a circle has infinitely many diameters, it has infinitely many lines of symmetry.
• Number of lines of symmetry = Infinite

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Question 4

Q: Copy the following figures on a squared paper and find the line(s) of symmetry.

For regular shapes, the lines of symmetry pass through:

• The midpoints of opposite sides (for polygons)
• The vertices and midpoints of opposite sides (for triangles)
• Every diameter (for circles)

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Question 5

Q: State whether the following statements are true or false:

(a) A square has 4 lines of symmetry. True. A square has 2 lines through midpoints of opposite sides and 2 diagonals, totalling 4 lines of symmetry.

(b) A rectangle has 4 lines of symmetry. False. A rectangle has only 2 lines of symmetry — one horizontal and one vertical. The diagonals are NOT lines of symmetry.

(c) A circle has infinite lines of symmetry. True. Every diameter of a circle is a line of symmetry, and a circle has infinitely many diameters.

(d) A parallelogram has a line of symmetry. False. A general parallelogram (that is not a rhombus or rectangle) has no line of symmetry.

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Exercise 13.2 — Lines of Symmetry in Figures, Letters, and Shapes

Question 1

Q: Find the number of lines of symmetry for each of the following shapes:

(a) A regular pentagon has 5 sides and 5 lines of symmetry. Each line passes through a vertex and the midpoint of the opposite side. Number of lines of symmetry = 5

(b) A regular hexagon has 6 sides and 6 lines of symmetry. Three lines pass through opposite vertices and three pass through midpoints of opposite sides. Number of lines of symmetry = 6

(c) A semicircle has exactly 1 line of symmetry — the line perpendicular to the base through its midpoint (the diameter). Number of lines of symmetry = 1

(d) An isosceles triangle has exactly 1 line of symmetry — the line from the vertex between the equal sides to the midpoint of the base. Number of lines of symmetry = 1

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Question 2

Q: Which of the following English alphabets have:

(a) Only one line of symmetry A, B, C, D, E, K, M, T, U, V, W, Y

These letters can be divided into identical halves by exactly one line.

• A: vertical line through the centre
• B: horizontal line through the centre
• C: horizontal line through the centre
• D: horizontal line through the centre
• E: horizontal line through the centre
• K: vertical or horizontal (depending on font, typically vertical)
• M: vertical line through the centre
• T: vertical line through the centre
• U: vertical line through the centre
• V: vertical line through the centre
• W: vertical line through the centre
• Y: vertical line through the centre

(b) Only two lines of symmetry H, I, X, O

• H: 1 vertical + 1 horizontal line
• I: 1 vertical + 1 horizontal line
• X: 1 vertical + 1 horizontal line (in some fonts, also diagonal)
• O: infinite lines (but in standard printed form often shown with 2)

(c) No line of symmetry F, G, J, L, N, P, Q, R, S, Z

These letters cannot be divided into two identical mirror halves by any line.

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Question 3

Q: Draw a rough sketch of a triangle with:

(a) Exactly one line of symmetry. An isosceles triangle has exactly one line of symmetry. The line passes from the vertex angle (between the two equal sides) to the midpoint of the base.

Steps to draw: 1. Draw a base of any length, say 6 cm. 2. Mark the midpoint of the base (3 cm from each end). 3. From this midpoint, draw a perpendicular line upward. 4. Mark a point on this perpendicular, say 5 cm above the base. 5. Join this point to both ends of the base. 6. The resulting triangle has two equal sides and one line of symmetry.

(b) Exactly three lines of symmetry. An equilateral triangle has exactly three lines of symmetry. Each line passes through one vertex and the midpoint of the opposite side.

Steps to draw: 1. Draw a line segment of 5 cm. 2. With each endpoint as centre, draw arcs of radius 5 cm. 3. Mark the intersection point of the arcs. 4. Join this point to both ends of the base. 5. The resulting equilateral triangle has 3 lines of symmetry.

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Question 4

Q: How many lines of symmetry does a regular polygon with n sides have?

For a regular polygon with n sides:

• Each line of symmetry passes through one vertex and the midpoint of the opposite side (if n is odd), or through opposite vertices / opposite midpoints (if n is even).
• The number of lines of symmetry equals the number of sides.

Number of lines of symmetry = n

Examples:

• Equilateral triangle (n = 3): 3 lines of symmetry
• Square (n = 4): 4 lines of symmetry
• Regular pentagon (n = 5): 5 lines of symmetry
• Regular hexagon (n = 6): 6 lines of symmetry

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Question 5

Q: Copy the figures with punched holes and find the axes of symmetry.

Step 1: Look at the figure and the position of the punched holes. Step 2: Try to find a line such that when the figure is folded along this line, the punched holes on one side coincide with the punched holes on the other side. Step 3: If such a line exists, it is a line of symmetry. Step 4: Check all possible lines — there may be more than one.

Key insight: The punched holes must be mirrored across the line of symmetry. If a hole is at a certain distance on one side of the line, there must be a corresponding hole at the same distance on the other side.

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Exercise 13.3 — Reflection and Mirror Images

Question 1

Q: Find the mirror image of the following figures along the mirror line shown.

Rules for mirror images:

• Points on the mirror line stay in the same position.
• Points far from the mirror line appear equally far on the opposite side.
• The mirror image is always the same size as the original.
• Left becomes right and right becomes left (lateral inversion).
• Top and bottom remain the same (no vertical inversion).

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Question 2

Q: Give the mirror image of the word "MATH".

Since M, A, T, and H are all vertically symmetrical letters, the mirror image of "MATH" looks the same as "MATH" when reflected in a vertical mirror.

However, when reflected in a horizontal mirror, each letter appears inverted top-to-bottom:

• M becomes W-like
• A becomes an inverted A
• T becomes an inverted T
• H stays H (horizontally symmetrical too)

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Question 3

Q: Complete the figure so that the mirror line is the line of symmetry.

Step 1: Take each point (vertex, corner) of the given part. Step 2: Measure its perpendicular distance from the mirror line. Step 3: Mark a point on the other side of the mirror line at the same perpendicular distance. Step 4: Join the reflected points in the same order as the original figure. Step 5: The completed figure should be symmetrical about the mirror line.

Example: If a right triangle is drawn to the left of a vertical mirror line with its hypotenuse sloping upward to the right, then the reflected right triangle on the right side will have its hypotenuse sloping upward to the left.

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Question 4

Q: Tell whether the following figures have reflection symmetry. If yes, how many lines of symmetry?

(a) A kite A kite has 1 line of symmetry — the line through the vertices where the equal sides meet. Yes, 1 line of symmetry.

(b) A parallelogram A general parallelogram has no line of symmetry. The opposite sides are equal, but the figure cannot be divided into two mirror halves by any line. No, 0 lines of symmetry.

(c) A rhombus A rhombus has 2 lines of symmetry — the two diagonals. Each diagonal divides the rhombus into two congruent triangles. Yes, 2 lines of symmetry.

(d) An isosceles trapezium An isosceles trapezium has 1 line of symmetry — the line through the midpoints of the two parallel sides. Yes, 1 line of symmetry.

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Question 5

Q: What letters of the English alphabet look the same when reflected in a vertical mirror?

A, H, I, M, O, T, U, V, W, X, Y

These letters have a vertical line of symmetry, so their left half is the mirror image of their right half. When placed in front of a vertical mirror, they appear unchanged.

Note: The letter O (in its circular form) has infinite lines of symmetry and will look the same in any mirror orientation.

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Exercise 13.4 — Applications and More on Symmetry

Question 1

Q: A figure has line symmetry if there is a line about which the figure may be folded so that the two parts coincide. Identify which of the following figures have line symmetry.

• Scalene triangle: No two sides are equal, so no fold line produces matching halves. No line of symmetry.
• Isosceles triangle: The line from the vertex between equal sides to the midpoint of the base divides it into two congruent halves. 1 line of symmetry.
• Square: Fold along horizontal, vertical, or diagonal lines — all produce matching halves. 4 lines of symmetry.
• Rhombus: The two diagonals are lines of symmetry. 2 lines of symmetry.
• Parallelogram (not rhombus, not rectangle): No line produces matching halves. 0 lines of symmetry.

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Question 2

Q: Which of the following numbers are symmetrical? Also find the number of lines of symmetry.

Note: The number of lines of symmetry depends on the font/style of the digit. In most standard printed fonts:

• 0 has 2 lines of symmetry (horizontal and vertical)
• 1 has no line of symmetry
• 3 has 1 line of symmetry (horizontal)
• 8 has 2 lines of symmetry (horizontal and vertical)

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Question 3

Q: Complete the following table:

• Equilateral Triangle (3): Three lines, each through a vertex and the midpoint of the opposite side.
• Isosceles Triangle (1): One line through the vertex angle and the midpoint of the base.
• Scalene Triangle (0): No two sides are equal, so no fold line works.
• Square (4): Two lines through midpoints of opposite sides + two diagonals.
• Rectangle (2): Only the lines through midpoints of opposite sides (diagonals are not lines of symmetry).
• Rhombus (2): The two diagonals are lines of symmetry.
• Parallelogram (0): No line produces mirror halves.
• Circle (Infinite): Every diameter is a line of symmetry.
• Semicircle (1): Only the line perpendicular to the diameter through its midpoint.
• Regular Pentagon (5): Five lines, each through a vertex and the midpoint of the opposite side.
• Regular Hexagon (6): Three lines through opposite vertices + three lines through midpoints of opposite sides.

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Question 4

Q: Draw the lines of symmetry for each of the following:

(a) An equilateral triangle

Steps: 1. Draw an equilateral triangle ABC with each side = 5 cm. 2. From vertex A, draw a line to the midpoint of side BC. This is one line of symmetry. 3. From vertex B, draw a line to the midpoint of side AC. This is the second line of symmetry. 4. From vertex C, draw a line to the midpoint of side AB. This is the third line of symmetry. 5. All three lines of symmetry meet at the centroid of the triangle.

(b) A regular hexagon

Steps: 1. Draw a regular hexagon with all sides equal and all angles equal. 2. Draw three lines through opposite vertices (these are 3 lines of symmetry). 3. Draw three lines through midpoints of opposite sides (these are the other 3 lines of symmetry). 4. All six lines of symmetry pass through the centre of the hexagon.

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Question 5

Q: Consider the letters of the word "ROTATION". Which letters have a line of symmetry? How many?

Letters with at least one line of symmetry: O, T, A, I Letters with no line of symmetry: R, N

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Common Mistakes

1. Confusing a diagonal with a line of symmetry in a rectangle. A rectangle has only 2 lines of symmetry (horizontal and vertical through the centre). The diagonals are NOT lines of symmetry — folding along a diagonal does not produce matching halves because the two triangles formed are not mirror images.

2. Assuming a parallelogram has a line of symmetry. A general parallelogram (not a rhombus or rectangle) has no line of symmetry. The diagonals do not divide it into mirror halves. Only rhombus and rectangle (special parallelograms) have lines of symmetry.

3. Counting the wrong number of lines for a regular polygon. A common mistake is to count only the lines through vertices and forget the lines through midpoints of opposite sides (or vice versa). Remember: a regular n-sided polygon always has exactly n lines of symmetry.

4. Thinking mirror image means just flipping left-right. A mirror image involves reflection across a specific line. The reflected point is at the same perpendicular distance from the mirror line on the opposite side. It is not simply reversing the figure; the position relative to the mirror line matters.

5. Forgetting that a circle has infinite lines of symmetry. Every diameter of a circle is a line of symmetry. Since there are infinitely many diameters, a circle has infinitely many lines of symmetry — not just 2 or 4.

6. Confusing reflection symmetry with rotational symmetry. Reflection symmetry involves a line (mirror line), while rotational symmetry involves rotation about a point by an angle less than 360 degrees. A figure can have one type without the other, or both.

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Exam Checklist

• [ ] I can define symmetry and identify a line of symmetry in a given figure.
• [ ] I can determine the number of lines of symmetry for common shapes (triangle types, quadrilateral types, circle, semicircle, regular polygons).
• [ ] I can distinguish between figures with 0, 1, 2, 3, 4, or infinite lines of symmetry.
• [ ] I can identify which English alphabet letters have vertical, horizontal, or no lines of symmetry.
• [ ] I can draw the mirror image of a figure across a given mirror line.
• [ ] I can complete a figure when one half and the mirror line are given.
• [ ] I can state the number of lines of symmetry for a regular polygon with n sides (answer: n).
• [ ] I can explain why a rectangle has only 2 lines of symmetry while a square has 4.
• [ ] I can explain why a parallelogram (not rhombus/rectangle) has no line of symmetry.
• [ ] I can apply the concept of symmetry to real-life objects (leaves, butterflies, architecture, etc.).

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