NCERT Solutions for Class 6 Maths Chapter 9: Symmetry — Free PDF

Chapter 9 of the NCERT Class 6 Maths textbook (2024-25) explores the beautiful concept of symmetry — one of the most fundamental ideas in both mathematics and nature. Symmetry is everywhere: in butterfly wings, flower petals, building facades, rangoli patterns, and even in the letters of the alphabet.

The key topics covered are:
- Line of symmetry (mirror line) — a line that divides a figure into two identical halves
- Reflection symmetry — when one half is the mirror image of the other
- Figures with multiple lines of symmetry — regular polygons, circles, and special shapes
- Rotational symmetry — when a figure looks the same after rotation by some angle
- Symmetry in everyday life — nature, art, architecture, and design

The chapter has 3 exercises that build from basic identification of symmetric figures to understanding rotational symmetry. This is a visual chapter that develops spatial reasoning and geometric intuition. Students who understand symmetry well will find it easier to work with geometry in higher classes.

Symmetry is not just a mathematical concept — it is a fundamental principle in physics (conservation laws), chemistry (molecular geometry), biology (bilateral symmetry in organisms), and art (design and architecture). Learning about symmetry at this stage builds a foundation that extends far beyond mathematics.

Symmetry: A figure is said to be symmetrical if it can be divided into two or more identical parts that are arranged in an organised pattern.

Line of symmetry (mirror line / axis of symmetry): A line that divides a figure into two parts such that each part is the mirror image of the other. If you fold the figure along this line, the two halves overlap perfectly.

Reflection symmetry: A figure has reflection symmetry if there exists at least one line of symmetry. The line acts like a mirror — every point on one side has a corresponding point at the same distance on the other side.

How to find a line of symmetry: Fold the figure along a line. If the two halves match exactly, that line is a line of symmetry. Alternatively, place a mirror along the line — the figure should look complete with its reflection.

Lines of symmetry for common shapes:
- Equilateral triangle: $3$ lines (each from a vertex to the midpoint of the opposite side)
- Isosceles triangle: $1$ line (from the vertex angle to the midpoint of the base)
- Scalene triangle: $0$ lines
- Square: $4$ lines ($2$ through midpoints of opposite sides + $2$ diagonals)
- Rectangle (non-square): $2$ lines (through midpoints of opposite sides; diagonals are NOT lines of symmetry)
- Rhombus (non-square): $2$ lines (both diagonals)
- Regular pentagon: $5$ lines
- Regular hexagon: $6$ lines
- **Regular $n$-gon:** $n$ lines
- Circle: Infinite lines (every diameter is a line of symmetry)

Reflection rules on the coordinate plane:
- Reflection across the $y$-axis: $(x, y) \rightarrow (-x, y)$
- Reflection across the $x$-axis: $(x, y) \rightarrow (x, -y)$

Rotational symmetry: A figure has rotational symmetry if it looks exactly the same after being rotated by some angle less than $360^\circ$ about a fixed point (centre of rotation).

Order of rotational symmetry: The number of times a figure looks the same during one complete rotation ($360^\circ$). A figure has rotational symmetry only if its order is $2$ or more.

Angle of rotation: $= \dfrac{360^\circ}{\text{order of rotational symmetry}}$

For example, a square has order $4$ and angle of rotation $= \dfrac{360^\circ}{4} = 90^\circ$.

Problem 1: How many lines of symmetry do the following shapes have: (a) equilateral triangle, (b) square, (c) rectangle, (d) circle?

Solution:

(a) Equilateral triangle: $3$ lines of symmetry — one from each vertex to the midpoint of the opposite side.

(b) Square: $4$ lines of symmetry — $2$ through opposite midpoints of sides and $2$ through opposite vertices (diagonals).

(c) Rectangle (non-square): $2$ lines of symmetry — one horizontal (through midpoints of longer sides) and one vertical (through midpoints of shorter sides). The diagonals are NOT lines of symmetry because folding along a diagonal does not make the halves overlap.

(d) Circle: Infinite lines of symmetry — every diameter is a line of symmetry.

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**Problem 2: How many lines of symmetry does a regular polygon with $n$ sides have?**

Solution:

A regular polygon has all sides equal and all angles equal.

Pattern: A regular polygon with $n$ sides has exactly $n$ lines of symmetry.

• If $n$ is odd, each line goes from a vertex to the midpoint of the opposite side.
  - If $n$ is even, half the lines go from vertex to vertex, and half go from midpoint to midpoint.

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Problem 3: Complete a symmetric figure given half of it and a line of symmetry.

Solution:

For each point on the given half:
1. Measure its perpendicular distance from the line of symmetry.
2. Mark a point on the other side at the same distance from the line.
3. Connect the reflected points in the same order.

Key rule: If a point is $d$ units from the line of symmetry, its mirror image is also $d$ units from the line, on the opposite side. The line of symmetry is the perpendicular bisector of the segment joining any point to its reflection.

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Problem 4: Does a parallelogram (that is not a rectangle or rhombus) have a line of symmetry?

Solution:

No. A general parallelogram has no line of symmetry. If you try folding along either diagonal or along lines through midpoints, the two halves do not overlap. However, a parallelogram does have rotational symmetry of order $2$ (it looks the same after a $180^\circ$ rotation).

**Problem 1: A triangle has vertices at $A(1, 3)$, $B(4, 3)$, $C(4, 1)$. Find its reflection across the $y$-axis.**

Solution:

When reflecting across the $y$-axis, the $x$-coordinate changes sign and the $y$-coordinate stays the same:

Answer: The reflected triangle has vertices $A'(-1, 3)$, $B'(-4, 3)$, $C'(-4, 1)$.

The reflected triangle has the same shape and size as the original — it is a mirror image.

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Problem 2: Which capital English letters have a horizontal line of symmetry? A vertical line of symmetry? Both?

Solution:

Vertical line of symmetry: $A, M, T, U, V, W, Y$

Horizontal line of symmetry: $B, C, D, E, K$

Both vertical and horizontal: $H, I, O, X$

No line of symmetry: $F, G, J, L, N, P, Q, R, S, Z$

Note that $O$ (when drawn as a perfect circle) actually has infinite lines of symmetry. Some words read the same in a mirror — for example, the word "MOM" has a vertical line of symmetry when written in block letters.

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**Problem 3: Reflect the point $(3, 5)$ across the $x$-axis.**

Solution:

When reflecting across the $x$-axis: $(x, y) \rightarrow (x, -y)$.

$(3, 5) \rightarrow (3, -5)$.

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Problem 4: A figure is drawn on one side of a mirror line. Describe how to complete the reflected figure using a ruler.

Solution:

For each key point (vertex) of the figure:
1. Draw a perpendicular from the point to the mirror line.
2. Extend the perpendicular an equal distance on the other side.
3. Mark the reflected point.
4. Join the reflected points in the same order as the original.

The resulting figure is the mirror image, and the mirror line is the perpendicular bisector of every segment joining a point to its reflection.

Problem 1: Does a square have rotational symmetry? If so, what is its order?

Solution:

Rotate a square about its centre:
- At $90^\circ$: the square looks the same ✓
- At $180^\circ$: the square looks the same ✓
- At $270^\circ$: the square looks the same ✓
- At $360^\circ$: back to original ✓

The square looks identical at $4$ positions during a full $360^\circ$ rotation.

Answer: Yes, a square has rotational symmetry of **order $4$**.

Angle of rotation $= \dfrac{360^\circ}{4} = 90^\circ$.

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Problem 2: Find the order of rotational symmetry for: (a) equilateral triangle, (b) regular hexagon, (c) circle, (d) rectangle.

Solution:

(a) Equilateral triangle: Looks the same at $120^\circ, 240^\circ, 360^\circ$ — order $= 3$.

(b) Regular hexagon: Looks the same at $60^\circ, 120^\circ, 180^\circ, 240^\circ, 300^\circ, 360^\circ$ — order $= 6$.

(c) Circle: Looks the same at any angle of rotation — order $= \infty$ (infinite).

(d) Rectangle: Looks the same at $180^\circ$ and $360^\circ$ — order $= 2$.

Pattern: A regular polygon with $n$ sides has rotational symmetry of order $n$.

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Problem 3: Can a shape have rotational symmetry but no line of symmetry?

Solution:

Yes! A parallelogram (that is not a rectangle or rhombus) has rotational symmetry of order $2$ (it looks the same after a $180^\circ$ rotation) but has no line of symmetry.

Other examples: the letter "S" and the letter "Z" both have rotational symmetry of order $2$ but no line of symmetry.

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**Problem 4: A fan has $3$ identical blades equally spaced. What is its order of rotational symmetry?**

Solution:

The $3$ blades are equally spaced, so the angle between consecutive blades is $\dfrac{360^\circ}{3} = 120^\circ$.

Rotating by $120^\circ$, $240^\circ$, or $360^\circ$ makes the fan look the same.

Answer: Order of rotational symmetry $= 3$.

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Problem 5: Does a rhombus have rotational symmetry? Does it have line symmetry?

Solution:

Rotational symmetry: Yes, order $2$ (the rhombus looks the same after a $180^\circ$ rotation).

Line symmetry: Yes, $2$ lines of symmetry — both diagonals.

So a rhombus has both types of symmetry.

**Example 1: A rangoli pattern has $8$ identical petals equally spaced around a centre. How many lines of symmetry does it have? What is its order of rotational symmetry?**

Solution:

Since there are $8$ identical petals equally spaced:
- Lines of symmetry $= 8$ (one through each petal, and one between each pair of adjacent petals)
- Order of rotational symmetry $= 8$
- Angle of rotation $= \dfrac{360^\circ}{8} = 45^\circ$

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**Example 2: A figure has exactly $1$ line of symmetry and no rotational symmetry. Give an example.**

Solution:

An isosceles triangle (that is not equilateral) has exactly $1$ line of symmetry (from the vertex angle to the midpoint of the base) and has rotational symmetry of order $1$ only (i.e., no rotational symmetry beyond the trivial $360^\circ$).

Another example: the letter "A" in block capitals.

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Example 3: How many lines of symmetry does a regular octagon have?

Solution:

A regular octagon has $8$ sides, so it has $8$ lines of symmetry:
- $4$ lines connecting midpoints of opposite sides
- $4$ lines connecting opposite vertices

Its order of rotational symmetry is also $8$, with angle of rotation $= \dfrac{360^\circ}{8} = 45^\circ$.

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Example 4: The word "BOOK" is written in block capital letters. Does the word as a whole have a line of symmetry?

Solution:

Looking at each letter: B has a horizontal line of symmetry, O has both, O has both, K has a horizontal line of symmetry.

Since all four letters have a horizontal line of symmetry, the entire word "BOOK" has a horizontal line of symmetry when written in appropriate block capitals.

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**Example 5: An arrow sign ($\rightarrow$) has how many lines of symmetry?**

Solution:

A standard arrow pointing right has $1$ line of symmetry — a horizontal line through its centre. It has no vertical line of symmetry (the pointed end and the tail end are different).

Mistake 1: Thinking rectangle diagonals are lines of symmetry.
This is a very common error. A rectangle (that is not a square) has only $2$ lines of symmetry, both passing through midpoints of opposite sides. The diagonals are NOT lines of symmetry because folding along a diagonal does not make the two halves coincide. In contrast, a square's diagonals ARE lines of symmetry.

Mistake 2: Confusing rotational symmetry with line symmetry.
A parallelogram has rotational symmetry (order $2$) but no line of symmetry. A shape can have one type without the other. Always check each type separately.

Mistake 3: Saying every shape has rotational symmetry.
Every shape returns to its original position after a $360^\circ$ rotation, but we only say it has rotational symmetry if the order is $2$ or more (i.e., it looks the same at some angle less than $360^\circ$).

Mistake 4: Miscounting lines of symmetry for irregular shapes.
Not every line through the centre of a shape is a line of symmetry. Each proposed line must be tested by the folding test — do the two halves match exactly?

Mistake 5: Assuming symmetry depends on the font for letters.
The symmetry of letters depends on how they are drawn. In exams, use standard block capital letters. For example, the letter "B" has a horizontal line of symmetry in block capitals but may not in a cursive or stylised font.

Q1. How many lines of symmetry does a regular hexagon have?

Answer: $6$ lines of symmetry ($3$ through opposite vertices + $3$ through midpoints of opposite sides).

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Q2. What is the order of rotational symmetry of the letter "H"?

Answer: Order $2$. The letter "H" looks the same after a $180^\circ$ rotation.

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Q3. A shape has rotational symmetry of order $5$. What is its angle of rotation?

Answer: Angle of rotation $= \dfrac{360^\circ}{5} = 72^\circ$.

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Q4. Does an isosceles trapezium have a line of symmetry?

Answer: Yes, an isosceles trapezium has $1$ line of symmetry — a vertical line through the midpoints of the two parallel sides.

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Q5. Give an example of a shape with exactly $2$ lines of symmetry and rotational symmetry of order $2$.

Answer: A rectangle (that is not a square) has $2$ lines of symmetry and rotational symmetry of order $2$. A rhombus (that is not a square) also satisfies this.

• A line of symmetry divides a figure into two mirror-image halves.
  - A regular polygon with $n$ sides has exactly $n$ lines of symmetry.
  - A circle has infinitely many lines of symmetry (every diameter).
  - Rotational symmetry means a figure looks the same after rotation by less than $360^\circ$.
  - Order of rotational symmetry is the number of times a figure matches itself during one full rotation. Angle $= \dfrac{360^\circ}{\text{order}}$.
  - A shape can have line symmetry without rotational symmetry (e.g., isosceles triangle), rotational symmetry without line symmetry (e.g., parallelogram), both (e.g., square), or neither (e.g., scalene triangle).
  - Reflection across the $y$-axis: $(x, y) \rightarrow (-x, y)$. Across the $x$-axis: $(x, y) \rightarrow (x, -y)$.
  - Rectangle diagonals are NOT lines of symmetry; square diagonals ARE.

Practise symmetry questions on SparkEd with $60$ interactive problems!