NCERT Textbook - Symmetry Class 7 Notes | EduRev

 Page 1


SYMMETRY 265 265 265 265 265
14.1  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.
Chapter  14
Symmetry
Nature
Architecture
Engineering
Create some colourful
Ink-dot devils
Make some symmetrical
paper-cut designs.
Compose a picture-album
 showing symmetry.
2020-21
©  NCERT 
not to be republished
Page 2


SYMMETRY 265 265 265 265 265
14.1  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.
Chapter  14
Symmetry
Nature
Architecture
Engineering
Create some colourful
Ink-dot devils
Make some symmetrical
paper-cut designs.
Compose a picture-album
 showing symmetry.
2020-21
©  NCERT 
not to be republished
MATHEMATICS 266 266 266 266 266
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 14.1 (i) to (iv)]
14.2  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 14.2).
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 14.3).
Fig 14.1
(i) (ii) (iii) ( i v)
60°
60°
60°
a a
a
Fig 14.2
Fig 14.3
2020-21
©  NCERT 
not to be republished
Page 3


SYMMETRY 265 265 265 265 265
14.1  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.
Chapter  14
Symmetry
Nature
Architecture
Engineering
Create some colourful
Ink-dot devils
Make some symmetrical
paper-cut designs.
Compose a picture-album
 showing symmetry.
2020-21
©  NCERT 
not to be republished
MATHEMATICS 266 266 266 266 266
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 14.1 (i) to (iv)]
14.2  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 14.2).
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 14.3).
Fig 14.1
(i) (ii) (iii) ( i v)
60°
60°
60°
a a
a
Fig 14.2
Fig 14.3
2020-21
©  NCERT 
not to be republished
SYMMETRY 267 267 267 267 267
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 14.4).
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 14.5).
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 14.6 (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 14.7). A mirror line,
thus, helps to visualise a line of symmetry (Fig 14.8).
Is the dotted line a mirror line? No. Is the dotted line a mirror line? Yes.
Fig 14.8
Fig 14.4
Fig 14.5
Fig 14.7
Fig 14.6
2020-21
©  NCERT 
not to be republished
Page 4


SYMMETRY 265 265 265 265 265
14.1  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.
Chapter  14
Symmetry
Nature
Architecture
Engineering
Create some colourful
Ink-dot devils
Make some symmetrical
paper-cut designs.
Compose a picture-album
 showing symmetry.
2020-21
©  NCERT 
not to be republished
MATHEMATICS 266 266 266 266 266
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 14.1 (i) to (iv)]
14.2  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 14.2).
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 14.3).
Fig 14.1
(i) (ii) (iii) ( i v)
60°
60°
60°
a a
a
Fig 14.2
Fig 14.3
2020-21
©  NCERT 
not to be republished
SYMMETRY 267 267 267 267 267
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 14.4).
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 14.5).
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 14.6 (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 14.7). A mirror line,
thus, helps to visualise a line of symmetry (Fig 14.8).
Is the dotted line a mirror line? No. Is the dotted line a mirror line? Yes.
Fig 14.8
Fig 14.4
Fig 14.5
Fig 14.7
Fig 14.6
2020-21
©  NCERT 
not to be republished
MATHEMATICS 268 268 268 268 268
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 14.9).
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 14.10
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 14.10).
EXERCISE 14.1
1. Copy the figures with punched holes and find the axes of symmetry for the following:
(i) (ii)
Fig 14.9
R R
2020-21
©  NCERT 
not to be republished
Page 5


SYMMETRY 265 265 265 265 265
14.1  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.
Chapter  14
Symmetry
Nature
Architecture
Engineering
Create some colourful
Ink-dot devils
Make some symmetrical
paper-cut designs.
Compose a picture-album
 showing symmetry.
2020-21
©  NCERT 
not to be republished
MATHEMATICS 266 266 266 266 266
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 14.1 (i) to (iv)]
14.2  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 14.2).
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 14.3).
Fig 14.1
(i) (ii) (iii) ( i v)
60°
60°
60°
a a
a
Fig 14.2
Fig 14.3
2020-21
©  NCERT 
not to be republished
SYMMETRY 267 267 267 267 267
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 14.4).
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 14.5).
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 14.6 (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 14.7). A mirror line,
thus, helps to visualise a line of symmetry (Fig 14.8).
Is the dotted line a mirror line? No. Is the dotted line a mirror line? Yes.
Fig 14.8
Fig 14.4
Fig 14.5
Fig 14.7
Fig 14.6
2020-21
©  NCERT 
not to be republished
MATHEMATICS 268 268 268 268 268
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 14.9).
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 14.10
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 14.10).
EXERCISE 14.1
1. Copy the figures with punched holes and find the axes of symmetry for the following:
(i) (ii)
Fig 14.9
R R
2020-21
©  NCERT 
not to be republished
SYMMETRY 269 269 269 269 269
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)
2020-21
©  NCERT 
not to be republished