NCERT Solutions for Class 7 Maths Chapter 12: Symmetry — Free PDF

A line of symmetry is a line that divides a figure into two identical halves that are mirror images of each other. If you fold the figure along this line, both halves overlap exactly.

Key facts:
- A figure can have $0$, $1$, $2$, or more lines of symmetry.
- A regular polygon with $n$ sides has exactly $n$ lines of symmetry.
- A circle has infinitely many lines of symmetry (every diameter is a line of symmetry).
- A scalene triangle has $0$ lines of symmetry.
- Lines of symmetry can be vertical, horizontal, or diagonal.

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 called the centre of rotation.

The smallest angle through which a figure can be rotated to look the same is called the angle of rotation. The formula is:

The order of rotational symmetry is the number of times a figure coincides with itself (looks exactly the same) during one complete rotation of $360^\circ$.

• If order $= 1$, the figure has no rotational symmetry (it only matches at $360^\circ$, which is trivial).
  - We say a figure has rotational symmetry only if the order is $\geq 2$.
  - A square has order $4$ (matches at $90^\circ$, $180^\circ$, $270^\circ$, $360^\circ$).
  - An equilateral triangle has order $3$ (matches at $120^\circ$, $240^\circ$, $360^\circ$).

Some figures have both line symmetry and rotational symmetry (e.g., a square), some have only one type (e.g., a parallelogram has rotational symmetry but no line symmetry), and some have neither (e.g., a scalene triangle).

For regular polygons, the number of lines of symmetry always equals the order of rotational symmetry. A regular $n$-sided polygon has $n$ lines and order $n$.

A regular hexagon has $6$ lines of symmetry — $3$ through opposite vertices and $3$ through midpoints of opposite sides.

General rule: A regular polygon with $n$ sides has $n$ lines of symmetry.

The letter A has $1$ line of symmetry — a vertical line through the middle. Letters F, G, and Z have no line of symmetry.

Note: O and X have $2$ lines of symmetry each (both vertical and horizontal).

A rhombus has $2$ lines of symmetry — both diagonals serve as lines of symmetry.

A rectangle has $2$ lines of symmetry — the lines joining the midpoints of opposite sides (not the diagonals). The diagonals of a rectangle are NOT lines of symmetry unless it is a square. This is because folding a rectangle along a diagonal does not make both halves overlap.

An arrow shape pointing right has $1$ line of symmetry (horizontal, through the middle). A plus sign (+) has $4$ lines of symmetry (horizontal, vertical, and two diagonal). A star with $5$ points (like a regular star) has $5$ lines of symmetry.

When checking for line symmetry, the practical test is: can you draw a line such that folding the figure along that line makes both halves coincide? If yes, that line is a line of symmetry.

A square looks the same at $90^\circ$, $180^\circ$, $270^\circ$, and $360^\circ$.

Order of rotational symmetry $= 4$.

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

An equilateral triangle looks the same at $120^\circ$, $240^\circ$, and $360^\circ$.

Order $= 3$. Angle of rotation $= \dfrac{360^\circ}{3} = 120^\circ$.

A rectangle (non-square) looks the same at $180^\circ$ and $360^\circ$.

Order $= 2$. Angle of rotation $= \dfrac{360^\circ}{2} = 180^\circ$.

A rectangle is an example of a figure that has both line symmetry ($2$ lines) and rotational symmetry (order $2$).

A scalene triangle only matches itself at $360^\circ$ (a full rotation). Its order is $1$, which means it has no rotational symmetry.

The letter S has rotational symmetry of order $2$ — it looks the same after a $180^\circ$ rotation. Interestingly, S has no line of symmetry. This makes S an example of a figure with rotational symmetry but no line symmetry.

A square has $4$ lines of symmetry AND rotational symmetry of order $4$. It is an example of a figure with both types of symmetry.

A parallelogram (non-rectangle, non-rhombus) has no line of symmetry, but it has rotational symmetry of order $2$ (it looks the same after a $180^\circ$ rotation). This is one of the most important exam facts from this chapter.

Yes, a figure can have rotational symmetry but no line symmetry. A parallelogram (that is not a rectangle or rhombus) has rotational symmetry of order $2$ but no line of symmetry. The letter S and the letter Z also have rotational symmetry of order $2$ but no line symmetry. The recycling symbol is another example.

Yes, a figure can have line symmetry but no rotational symmetry. An isosceles triangle (that is not equilateral) has $1$ line of symmetry but only has trivial rotational symmetry (order $1$). A kite also has $1$ line of symmetry but no rotational symmetry.

So the two types of symmetry are independent — having one does not guarantee the other.

For a regular polygon with $n$ sides, the number of lines of symmetry equals $n$ and the order of rotational symmetry also equals $n$. They are always the same.

Examples: equilateral triangle ($3$ lines, order $3$), square ($4$ lines, order $4$), regular pentagon ($5$ lines, order $5$), regular hexagon ($6$ lines, order $6$).

A figure has rotational symmetry of order $6$. What is the angle of rotation?

Solution:

The figure looks the same after every $60^\circ$ of rotation. It coincides with itself at $60^\circ$, $120^\circ$, $180^\circ$, $240^\circ$, $300^\circ$, and $360^\circ$.

How many lines of symmetry does a regular octagon ($8$ sides) have? What is its order of rotational symmetry?

Solution:
A regular octagon has $8$ lines of symmetry ($4$ through opposite vertices and $4$ through midpoints of opposite sides).

Order of rotational symmetry $= 8$.
Angle of rotation $= \dfrac{360^\circ}{8} = 45^\circ$.

Name a quadrilateral that has: (a) exactly $2$ lines of symmetry (b) no lines of symmetry but rotational symmetry of order $2$.

Solution:
(a) A rectangle (non-square) or a rhombus (non-square) — both have exactly $2$ lines of symmetry.
(b) A parallelogram (that is neither a rectangle nor a rhombus) — it has no line of symmetry but has rotational symmetry of order $2$.

A windmill has $4$ blades equally spaced. What is the order of rotational symmetry? Does it have line symmetry?

Solution:
Order of rotational symmetry $= 4$ (it looks the same after $90^\circ$, $180^\circ$, $270^\circ$, and $360^\circ$ rotations).

If the blades are symmetric in shape, it has $4$ lines of symmetry. If each blade is curved (like a real windmill), it may have rotational symmetry but no line symmetry.

The digit $8$ has how many lines of symmetry? What is its order of rotational symmetry?

Solution:
The digit $8$ has $2$ lines of symmetry — one vertical and one horizontal.
It has rotational symmetry of order $2$ (looks the same after $180^\circ$ rotation).

The diagonals of a rectangle are NOT lines of symmetry (unless it is a square). A line of symmetry must divide the figure into two mirror-image halves. The diagonals of a rectangle create two triangles that are congruent but are NOT mirror images across the diagonal.

A general parallelogram has NO line of symmetry. This is a very commonly tested fact. Only special parallelograms (rectangles, rhombi, squares) have line symmetry. A parallelogram does have rotational symmetry of order $2$.

Order $=$ how many times the figure matches itself during a full turn. Angle $= \dfrac{360^\circ}{\text{order}}$. If order $= 4$, angle $= 90^\circ$ (not $4^\circ$). These are inversely related — higher order means smaller angle.

Every figure matches itself after a $360^\circ$ rotation, so every figure has order at least $1$. We say a figure has rotational symmetry only when the order is $2$ or more. A scalene triangle has order $1$ — this does not count as rotational symmetry.

The letter O (as a perfect oval shape) has $2$ lines of symmetry (vertical and horizontal). If treated as a circle, it has infinitely many. The answer depends on how the letter is drawn — check the context of the question.

A regular polygon has rotational symmetry of order $10$. How many sides does it have? What is its angle of rotation?

Answer: It has $10$ sides (regular decagon). Angle of rotation $= \dfrac{360^\circ}{10} = 36^\circ$.

For each shape, state whether it has (i) line symmetry only, (ii) rotational symmetry only, (iii) both, or (iv) neither: (a) Isosceles triangle (b) Parallelogram (c) Square (d) Scalene triangle

Answer:
(a) Line symmetry only ($1$ line, order $1$)
(b) Rotational symmetry only ($0$ lines, order $2$)
(c) Both ($4$ lines, order $4$)
(d) Neither ($0$ lines, order $1$)

Which English capital letters have rotational symmetry of order $2$?

Answer: H, I, N, O, S, X, Z all have rotational symmetry of order $2$ — they look the same after a $180^\circ$ rotation.

A rangoli pattern has $8$ lines of symmetry and rotational symmetry of order $8$. What is its angle of rotation? Name a regular polygon with the same symmetry properties.

Answer: Angle of rotation $= \dfrac{360^\circ}{8} = 45^\circ$. A regular octagon has $8$ lines of symmetry and order $8$ rotational symmetry.

(a) Every square is a rhombus, so a square has at least $2$ lines of symmetry. — True (a square has $4$ lines).
(b) A circle has exactly $360$ lines of symmetry. — False (a circle has infinitely many).
(c) The letter N has one line of symmetry. — False (N has no line of symmetry, but it has rotational symmetry of order $2$).
(d) A regular hexagon has $3$ lines of symmetry. — False (it has $6$ lines of symmetry).