NCERT Textbook: Symmetry | NCERT Textbooks (Class 6 to Class 12) - CTET & State TET PDF Download

 Page 1


MATHEMATICS 186
12.1  INTRODUCTION
Symmetry is an important geometrical concept, commonly exhibited in nature and is used
almost in every field of activity.  Artists, professionals, designers of clothing or jewellery,
car manufacturers, architects and many others make use of the idea of symmetry.  The
beehives, the flowers, the tree-leaves, religious symbols, rugs, and handkerchiefs —
everywhere you find symmetrical designs.
Y ou have already had a ‘feel’ of line symmetry in your previous class.
A figure has a line symmetry , if there is a line about which the figure may be folded so
that the two parts of the figure will coincide.
Y ou might like to recall these ideas.  Here are some activities to help you.
Nature
Architecture
Engineering
Create some colourful
Ink-dot devils
Make some symmetrical
paper-cut designs.
Compose a picture-album
 showing symmetry.
Chapter  12
Symmetry
2024-25
Page 2


MATHEMATICS 186
12.1  INTRODUCTION
Symmetry is an important geometrical concept, commonly exhibited in nature and is used
almost in every field of activity.  Artists, professionals, designers of clothing or jewellery,
car manufacturers, architects and many others make use of the idea of symmetry.  The
beehives, the flowers, the tree-leaves, religious symbols, rugs, and handkerchiefs —
everywhere you find symmetrical designs.
Y ou have already had a ‘feel’ of line symmetry in your previous class.
A figure has a line symmetry , if there is a line about which the figure may be folded so
that the two parts of the figure will coincide.
Y ou might like to recall these ideas.  Here are some activities to help you.
Nature
Architecture
Engineering
Create some colourful
Ink-dot devils
Make some symmetrical
paper-cut designs.
Compose a picture-album
 showing symmetry.
Chapter  12
Symmetry
2024-25
SYMMETRY 187
Enjoy identifying lines (also called axes) of symmetry in the designs you collect.
Let us now strengthen our ideas on symmetry further.  Study the following figures in
which the lines of symmetry are marked with dotted lines. [Fig 12.1 (i) to (iv)]
12.2  LINES OF SYMMETRY FOR REGULAR POLYGONS
Y ou know that a polygon is a closed figure made of several line segments.  The polygon
made up of the least number of line segments is the triangle. (Can there be a polygon that
you can draw with still fewer line segments? Think about it).
A polygon is said to be regular if all its sides are of equal length and all its angles are of
equal measure. Thus, an equilateral triangle is a regular polygon of three sides. Can you
name the regular polygon of four sides?
An equilateral triangle is regular because each of its sides has same length and each of
its angles measures 60° (Fig 12.2).
A square is also regular because all its sides are of equal length and each of its angles
is a right angle (i.e., 90°).  Its diagonals are seen to be perpendicular bisectors of one
another (Fig 12.3).
Fig 12.1
(i) (ii) (iii) (iv)
60°
60°
60°
a a
a
Fig 12.2
Fig 12.3
2024-25
Page 3


MATHEMATICS 186
12.1  INTRODUCTION
Symmetry is an important geometrical concept, commonly exhibited in nature and is used
almost in every field of activity.  Artists, professionals, designers of clothing or jewellery,
car manufacturers, architects and many others make use of the idea of symmetry.  The
beehives, the flowers, the tree-leaves, religious symbols, rugs, and handkerchiefs —
everywhere you find symmetrical designs.
Y ou have already had a ‘feel’ of line symmetry in your previous class.
A figure has a line symmetry , if there is a line about which the figure may be folded so
that the two parts of the figure will coincide.
Y ou might like to recall these ideas.  Here are some activities to help you.
Nature
Architecture
Engineering
Create some colourful
Ink-dot devils
Make some symmetrical
paper-cut designs.
Compose a picture-album
 showing symmetry.
Chapter  12
Symmetry
2024-25
SYMMETRY 187
Enjoy identifying lines (also called axes) of symmetry in the designs you collect.
Let us now strengthen our ideas on symmetry further.  Study the following figures in
which the lines of symmetry are marked with dotted lines. [Fig 12.1 (i) to (iv)]
12.2  LINES OF SYMMETRY FOR REGULAR POLYGONS
Y ou know that a polygon is a closed figure made of several line segments.  The polygon
made up of the least number of line segments is the triangle. (Can there be a polygon that
you can draw with still fewer line segments? Think about it).
A polygon is said to be regular if all its sides are of equal length and all its angles are of
equal measure. Thus, an equilateral triangle is a regular polygon of three sides. Can you
name the regular polygon of four sides?
An equilateral triangle is regular because each of its sides has same length and each of
its angles measures 60° (Fig 12.2).
A square is also regular because all its sides are of equal length and each of its angles
is a right angle (i.e., 90°).  Its diagonals are seen to be perpendicular bisectors of one
another (Fig 12.3).
Fig 12.1
(i) (ii) (iii) (iv)
60°
60°
60°
a a
a
Fig 12.2
Fig 12.3
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MATHEMATICS 188
If a pentagon is regular, naturally , its sides should have equal length. Y ou will, later on,
learn that the measure of each of its  angles is 108° (Fig 12.4).
A regular hexagon has all its sides equal and each of its angles measures 120°. Y ou will
learn more of these figures later (Fig 12.5).
The regular polygons are symmetrical figures and hence their lines of symmetry are
quite interesting,
Each regular polygon has as many lines of symmetry as it has sides [Fig 12.6 (i) - (iv)].
W e say, they have multiple lines of symmetry.
Perhaps, you might like to investigate this by paper folding. Go ahead!
The concept of line symmetry is closely related to mirror reflection.  A shape has line
symmetry when one half of it is the mirror image of the other half (Fig 12.7). A mirror line,
thus, helps to visualise a line of symmetry (Fig 12.8).
Is the dotted line a mirror line? No. Is the dotted line a mirror line? Yes.
Fig 12.8
Fig 12.4
Fig 12.6
Fig 12.7
Fig 12.5
2024-25
Page 4


MATHEMATICS 186
12.1  INTRODUCTION
Symmetry is an important geometrical concept, commonly exhibited in nature and is used
almost in every field of activity.  Artists, professionals, designers of clothing or jewellery,
car manufacturers, architects and many others make use of the idea of symmetry.  The
beehives, the flowers, the tree-leaves, religious symbols, rugs, and handkerchiefs —
everywhere you find symmetrical designs.
Y ou have already had a ‘feel’ of line symmetry in your previous class.
A figure has a line symmetry , if there is a line about which the figure may be folded so
that the two parts of the figure will coincide.
Y ou might like to recall these ideas.  Here are some activities to help you.
Nature
Architecture
Engineering
Create some colourful
Ink-dot devils
Make some symmetrical
paper-cut designs.
Compose a picture-album
 showing symmetry.
Chapter  12
Symmetry
2024-25
SYMMETRY 187
Enjoy identifying lines (also called axes) of symmetry in the designs you collect.
Let us now strengthen our ideas on symmetry further.  Study the following figures in
which the lines of symmetry are marked with dotted lines. [Fig 12.1 (i) to (iv)]
12.2  LINES OF SYMMETRY FOR REGULAR POLYGONS
Y ou know that a polygon is a closed figure made of several line segments.  The polygon
made up of the least number of line segments is the triangle. (Can there be a polygon that
you can draw with still fewer line segments? Think about it).
A polygon is said to be regular if all its sides are of equal length and all its angles are of
equal measure. Thus, an equilateral triangle is a regular polygon of three sides. Can you
name the regular polygon of four sides?
An equilateral triangle is regular because each of its sides has same length and each of
its angles measures 60° (Fig 12.2).
A square is also regular because all its sides are of equal length and each of its angles
is a right angle (i.e., 90°).  Its diagonals are seen to be perpendicular bisectors of one
another (Fig 12.3).
Fig 12.1
(i) (ii) (iii) (iv)
60°
60°
60°
a a
a
Fig 12.2
Fig 12.3
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MATHEMATICS 188
If a pentagon is regular, naturally , its sides should have equal length. Y ou will, later on,
learn that the measure of each of its  angles is 108° (Fig 12.4).
A regular hexagon has all its sides equal and each of its angles measures 120°. Y ou will
learn more of these figures later (Fig 12.5).
The regular polygons are symmetrical figures and hence their lines of symmetry are
quite interesting,
Each regular polygon has as many lines of symmetry as it has sides [Fig 12.6 (i) - (iv)].
W e say, they have multiple lines of symmetry.
Perhaps, you might like to investigate this by paper folding. Go ahead!
The concept of line symmetry is closely related to mirror reflection.  A shape has line
symmetry when one half of it is the mirror image of the other half (Fig 12.7). A mirror line,
thus, helps to visualise a line of symmetry (Fig 12.8).
Is the dotted line a mirror line? No. Is the dotted line a mirror line? Yes.
Fig 12.8
Fig 12.4
Fig 12.6
Fig 12.7
Fig 12.5
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SYMMETRY 189
While dealing with mirror reflection, care is needed to note down the left-right changes
in the orientation, as seen in the figure here (Fig 12.9).
The shape is same, but the other way round!
Play this punching game!
Fold a sheet into two halves Punch a hole two holes about the
symmetric fold.
Fig 12.10
The fold is a line (or axis) of symmetry .  Study about punches at different locations on
the folded paper and the corresponding lines of symmetry (Fig 12.10).
EXERCISE 12.1
1. Copy the figures with punched holes and find the axes of symmetry for the following:
Fig 12.9
(i)
R R
(ii)
2024-25
Page 5


MATHEMATICS 186
12.1  INTRODUCTION
Symmetry is an important geometrical concept, commonly exhibited in nature and is used
almost in every field of activity.  Artists, professionals, designers of clothing or jewellery,
car manufacturers, architects and many others make use of the idea of symmetry.  The
beehives, the flowers, the tree-leaves, religious symbols, rugs, and handkerchiefs —
everywhere you find symmetrical designs.
Y ou have already had a ‘feel’ of line symmetry in your previous class.
A figure has a line symmetry , if there is a line about which the figure may be folded so
that the two parts of the figure will coincide.
Y ou might like to recall these ideas.  Here are some activities to help you.
Nature
Architecture
Engineering
Create some colourful
Ink-dot devils
Make some symmetrical
paper-cut designs.
Compose a picture-album
 showing symmetry.
Chapter  12
Symmetry
2024-25
SYMMETRY 187
Enjoy identifying lines (also called axes) of symmetry in the designs you collect.
Let us now strengthen our ideas on symmetry further.  Study the following figures in
which the lines of symmetry are marked with dotted lines. [Fig 12.1 (i) to (iv)]
12.2  LINES OF SYMMETRY FOR REGULAR POLYGONS
Y ou know that a polygon is a closed figure made of several line segments.  The polygon
made up of the least number of line segments is the triangle. (Can there be a polygon that
you can draw with still fewer line segments? Think about it).
A polygon is said to be regular if all its sides are of equal length and all its angles are of
equal measure. Thus, an equilateral triangle is a regular polygon of three sides. Can you
name the regular polygon of four sides?
An equilateral triangle is regular because each of its sides has same length and each of
its angles measures 60° (Fig 12.2).
A square is also regular because all its sides are of equal length and each of its angles
is a right angle (i.e., 90°).  Its diagonals are seen to be perpendicular bisectors of one
another (Fig 12.3).
Fig 12.1
(i) (ii) (iii) (iv)
60°
60°
60°
a a
a
Fig 12.2
Fig 12.3
2024-25
MATHEMATICS 188
If a pentagon is regular, naturally , its sides should have equal length. Y ou will, later on,
learn that the measure of each of its  angles is 108° (Fig 12.4).
A regular hexagon has all its sides equal and each of its angles measures 120°. Y ou will
learn more of these figures later (Fig 12.5).
The regular polygons are symmetrical figures and hence their lines of symmetry are
quite interesting,
Each regular polygon has as many lines of symmetry as it has sides [Fig 12.6 (i) - (iv)].
W e say, they have multiple lines of symmetry.
Perhaps, you might like to investigate this by paper folding. Go ahead!
The concept of line symmetry is closely related to mirror reflection.  A shape has line
symmetry when one half of it is the mirror image of the other half (Fig 12.7). A mirror line,
thus, helps to visualise a line of symmetry (Fig 12.8).
Is the dotted line a mirror line? No. Is the dotted line a mirror line? Yes.
Fig 12.8
Fig 12.4
Fig 12.6
Fig 12.7
Fig 12.5
2024-25
SYMMETRY 189
While dealing with mirror reflection, care is needed to note down the left-right changes
in the orientation, as seen in the figure here (Fig 12.9).
The shape is same, but the other way round!
Play this punching game!
Fold a sheet into two halves Punch a hole two holes about the
symmetric fold.
Fig 12.10
The fold is a line (or axis) of symmetry .  Study about punches at different locations on
the folded paper and the corresponding lines of symmetry (Fig 12.10).
EXERCISE 12.1
1. Copy the figures with punched holes and find the axes of symmetry for the following:
Fig 12.9
(i)
R R
(ii)
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MATHEMATICS 190
2. Given the line(s) of symmetry , find the other hole(s):
3. In the following figures, the mirror line (i.e., the line of symmetry) is given as a dotted
line. Complete each figure performing reflection in the dotted (mirror) line. (Y ou might
perhaps place a mirror along the dotted line and look into the mirror for the image).
Are you able to recall the name of the figure you complete?
 4. The following figures have more than one line of symmetry.  Such figures are said to
have multiple lines of symmetry .
Identify multiple lines of symmetry , if any , in each of the following figures:
(a) (b) (c)
(a) (b) (c) (d) (e) (f)
2024-25