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

 Page 1


Symmetry is quite a common term used in day to day life. When we see
certain figures with evenly balanced proportions, we say, “They are
symmetrical”.
These pictures of architectural marvel are beautiful
because of their symmetry.
Suppose we could fold a picture in half such that the
left and right halves match exactly then the picture is said
to have line symmetry (Fig 13.1). We can see that the two
halves are mirror images of each other. If we place a mirror
on the fold then the image of one side of the picture will
fall exactly on the other side of the picture. When it happens,
the fold, which is the mirror line, is a line of symmetry (or
an axis of symmetry) for the picture.
Tajmahal (U.P.) Thiruvannamalai (Tamil Nadu)
Fig 13.1
13.1 Introduction
Chapter 13 Chapter 13 Chapter 13 Chapter 13 Chapter 13
Symmetry Symmetry
Symmetry Symmetry Symmetry
2022-23
Page 2


Symmetry is quite a common term used in day to day life. When we see
certain figures with evenly balanced proportions, we say, “They are
symmetrical”.
These pictures of architectural marvel are beautiful
because of their symmetry.
Suppose we could fold a picture in half such that the
left and right halves match exactly then the picture is said
to have line symmetry (Fig 13.1). We can see that the two
halves are mirror images of each other. If we place a mirror
on the fold then the image of one side of the picture will
fall exactly on the other side of the picture. When it happens,
the fold, which is the mirror line, is a line of symmetry (or
an axis of symmetry) for the picture.
Tajmahal (U.P.) Thiruvannamalai (Tamil Nadu)
Fig 13.1
13.1 Introduction
Chapter 13 Chapter 13 Chapter 13 Chapter 13 Chapter 13
Symmetry Symmetry
Symmetry Symmetry Symmetry
2022-23
MATHEMATICS
262
The shapes you see here are symmetrical. Why?
When you fold them along the dotted line, one half of
the drawing would fit exactly over the other half.
How do you name the dotted line in the figure 13.1?
Where will you place the mirror for having the image
exactly over the other half of the picture?
The adjacent figure 13.2 is not symmetrical.
Can you tell ‘why not’?
13.2 Making Symmetric Figures : Ink-blot Devils
Take a piece of paper. Fold it in half.
Spill a few drops of ink on one half side.
Now press the halves together.
What do you see?
Is the resulting figure symmetric? If yes, where is
the line of symmetry? Is there any other line along which
it can be folded to produce two identical parts?
Try more such patterns.
Inked-string patterns
Fold a paper in half. On one half-portion,  arrange short lengths of string
dipped in a variety of coloured inks or paints. Now press the two halves.
Study the figure you obtain. Is it symmetric? In how many ways can it be
folded to produce two identical halves?
List a few objects you find in your class
room such as the black board, the table, the
wall, the textbook, etc. Which of them are
symmetric and which are not? Can you identify
the lines of symmetry for those objects which
are symmetric?
Do This
Fig 13.2
You have two set-squares
in your ‘mathematical
instruments box’. Are they
symmetric?
2022-23
Page 3


Symmetry is quite a common term used in day to day life. When we see
certain figures with evenly balanced proportions, we say, “They are
symmetrical”.
These pictures of architectural marvel are beautiful
because of their symmetry.
Suppose we could fold a picture in half such that the
left and right halves match exactly then the picture is said
to have line symmetry (Fig 13.1). We can see that the two
halves are mirror images of each other. If we place a mirror
on the fold then the image of one side of the picture will
fall exactly on the other side of the picture. When it happens,
the fold, which is the mirror line, is a line of symmetry (or
an axis of symmetry) for the picture.
Tajmahal (U.P.) Thiruvannamalai (Tamil Nadu)
Fig 13.1
13.1 Introduction
Chapter 13 Chapter 13 Chapter 13 Chapter 13 Chapter 13
Symmetry Symmetry
Symmetry Symmetry Symmetry
2022-23
MATHEMATICS
262
The shapes you see here are symmetrical. Why?
When you fold them along the dotted line, one half of
the drawing would fit exactly over the other half.
How do you name the dotted line in the figure 13.1?
Where will you place the mirror for having the image
exactly over the other half of the picture?
The adjacent figure 13.2 is not symmetrical.
Can you tell ‘why not’?
13.2 Making Symmetric Figures : Ink-blot Devils
Take a piece of paper. Fold it in half.
Spill a few drops of ink on one half side.
Now press the halves together.
What do you see?
Is the resulting figure symmetric? If yes, where is
the line of symmetry? Is there any other line along which
it can be folded to produce two identical parts?
Try more such patterns.
Inked-string patterns
Fold a paper in half. On one half-portion,  arrange short lengths of string
dipped in a variety of coloured inks or paints. Now press the two halves.
Study the figure you obtain. Is it symmetric? In how many ways can it be
folded to produce two identical halves?
List a few objects you find in your class
room such as the black board, the table, the
wall, the textbook, etc. Which of them are
symmetric and which are not? Can you identify
the lines of symmetry for those objects which
are symmetric?
Do This
Fig 13.2
You have two set-squares
in your ‘mathematical
instruments box’. Are they
symmetric?
2022-23
SYMMETRY
263
EXERCISE 13.1
1. List any four symmetrical objects from your home or school.
2. For the given figure, which one is the mirror line, l
1
 or l
2
?
3. Identify the shapes given below. Check whether they are
symmetric or not. Draw the line of symmetry as well.
4. Copy the following on a squared paper. A square paper is what you would have used
in your arithmetic notebook in earlier classes. Then complete them such that the dotted
line is the line of symmetry .
(a) (b) (c)
(d) (e) (f )
(a) (b) (c)
(d) (e) (f)
l
l
2
l
1
5. In the figure, l is the line of symmetry .
Complete the diagram to make it symmetric.
2022-23
Page 4


Symmetry is quite a common term used in day to day life. When we see
certain figures with evenly balanced proportions, we say, “They are
symmetrical”.
These pictures of architectural marvel are beautiful
because of their symmetry.
Suppose we could fold a picture in half such that the
left and right halves match exactly then the picture is said
to have line symmetry (Fig 13.1). We can see that the two
halves are mirror images of each other. If we place a mirror
on the fold then the image of one side of the picture will
fall exactly on the other side of the picture. When it happens,
the fold, which is the mirror line, is a line of symmetry (or
an axis of symmetry) for the picture.
Tajmahal (U.P.) Thiruvannamalai (Tamil Nadu)
Fig 13.1
13.1 Introduction
Chapter 13 Chapter 13 Chapter 13 Chapter 13 Chapter 13
Symmetry Symmetry
Symmetry Symmetry Symmetry
2022-23
MATHEMATICS
262
The shapes you see here are symmetrical. Why?
When you fold them along the dotted line, one half of
the drawing would fit exactly over the other half.
How do you name the dotted line in the figure 13.1?
Where will you place the mirror for having the image
exactly over the other half of the picture?
The adjacent figure 13.2 is not symmetrical.
Can you tell ‘why not’?
13.2 Making Symmetric Figures : Ink-blot Devils
Take a piece of paper. Fold it in half.
Spill a few drops of ink on one half side.
Now press the halves together.
What do you see?
Is the resulting figure symmetric? If yes, where is
the line of symmetry? Is there any other line along which
it can be folded to produce two identical parts?
Try more such patterns.
Inked-string patterns
Fold a paper in half. On one half-portion,  arrange short lengths of string
dipped in a variety of coloured inks or paints. Now press the two halves.
Study the figure you obtain. Is it symmetric? In how many ways can it be
folded to produce two identical halves?
List a few objects you find in your class
room such as the black board, the table, the
wall, the textbook, etc. Which of them are
symmetric and which are not? Can you identify
the lines of symmetry for those objects which
are symmetric?
Do This
Fig 13.2
You have two set-squares
in your ‘mathematical
instruments box’. Are they
symmetric?
2022-23
SYMMETRY
263
EXERCISE 13.1
1. List any four symmetrical objects from your home or school.
2. For the given figure, which one is the mirror line, l
1
 or l
2
?
3. Identify the shapes given below. Check whether they are
symmetric or not. Draw the line of symmetry as well.
4. Copy the following on a squared paper. A square paper is what you would have used
in your arithmetic notebook in earlier classes. Then complete them such that the dotted
line is the line of symmetry .
(a) (b) (c)
(d) (e) (f )
(a) (b) (c)
(d) (e) (f)
l
l
2
l
1
5. In the figure, l is the line of symmetry .
Complete the diagram to make it symmetric.
2022-23
MATHEMATICS
264
6. In the figure, l is the line of symmetry.
Draw the image of the triangle and complete the  diagram
so that it becomes symmetric.
13.3 Figures with Two Lines of Symmetry
A kite
One of the two set-squares in your instrument box has angles of measure 30°,
60°, 90°.
Take two such identical set-squares. Place them side by side
to form a ‘kite’, like the one shown here.
How many lines of symmetry does the shape have?
Do you think that some shapes may have more than one line
of symmetry?
A rectangle
Take a rectangular sheet (like a post-card). Fold it once lengthwise so that one
half fits exactly over the other half. Is this fold a line of symmetry? Why?
Open it up now and
again fold on its
width in the same
way. Is this second
fold also a line of
symmetry? Why?
Do you find that these two lines are the lines of
symmetry?
A cut out from double fold
Take a rectangular piece of paper. Fold
it once and then once more. Draw
some design as shown. Cut the shape
drawn and unfold the shape. (Before
unfolding, try to guess the shape you
are likely to get).
How many lines of symmetry
does the shape have  which has been
cut out?
Create more such designs.
Do This
1st fold
 2nd fold
Form as many
shapes as you
can by
combining two
or more set
squares.  Draw
them on squared
paper and note
their lines of
symmetry.
l
2022-23
Page 5


Symmetry is quite a common term used in day to day life. When we see
certain figures with evenly balanced proportions, we say, “They are
symmetrical”.
These pictures of architectural marvel are beautiful
because of their symmetry.
Suppose we could fold a picture in half such that the
left and right halves match exactly then the picture is said
to have line symmetry (Fig 13.1). We can see that the two
halves are mirror images of each other. If we place a mirror
on the fold then the image of one side of the picture will
fall exactly on the other side of the picture. When it happens,
the fold, which is the mirror line, is a line of symmetry (or
an axis of symmetry) for the picture.
Tajmahal (U.P.) Thiruvannamalai (Tamil Nadu)
Fig 13.1
13.1 Introduction
Chapter 13 Chapter 13 Chapter 13 Chapter 13 Chapter 13
Symmetry Symmetry
Symmetry Symmetry Symmetry
2022-23
MATHEMATICS
262
The shapes you see here are symmetrical. Why?
When you fold them along the dotted line, one half of
the drawing would fit exactly over the other half.
How do you name the dotted line in the figure 13.1?
Where will you place the mirror for having the image
exactly over the other half of the picture?
The adjacent figure 13.2 is not symmetrical.
Can you tell ‘why not’?
13.2 Making Symmetric Figures : Ink-blot Devils
Take a piece of paper. Fold it in half.
Spill a few drops of ink on one half side.
Now press the halves together.
What do you see?
Is the resulting figure symmetric? If yes, where is
the line of symmetry? Is there any other line along which
it can be folded to produce two identical parts?
Try more such patterns.
Inked-string patterns
Fold a paper in half. On one half-portion,  arrange short lengths of string
dipped in a variety of coloured inks or paints. Now press the two halves.
Study the figure you obtain. Is it symmetric? In how many ways can it be
folded to produce two identical halves?
List a few objects you find in your class
room such as the black board, the table, the
wall, the textbook, etc. Which of them are
symmetric and which are not? Can you identify
the lines of symmetry for those objects which
are symmetric?
Do This
Fig 13.2
You have two set-squares
in your ‘mathematical
instruments box’. Are they
symmetric?
2022-23
SYMMETRY
263
EXERCISE 13.1
1. List any four symmetrical objects from your home or school.
2. For the given figure, which one is the mirror line, l
1
 or l
2
?
3. Identify the shapes given below. Check whether they are
symmetric or not. Draw the line of symmetry as well.
4. Copy the following on a squared paper. A square paper is what you would have used
in your arithmetic notebook in earlier classes. Then complete them such that the dotted
line is the line of symmetry .
(a) (b) (c)
(d) (e) (f )
(a) (b) (c)
(d) (e) (f)
l
l
2
l
1
5. In the figure, l is the line of symmetry .
Complete the diagram to make it symmetric.
2022-23
MATHEMATICS
264
6. In the figure, l is the line of symmetry.
Draw the image of the triangle and complete the  diagram
so that it becomes symmetric.
13.3 Figures with Two Lines of Symmetry
A kite
One of the two set-squares in your instrument box has angles of measure 30°,
60°, 90°.
Take two such identical set-squares. Place them side by side
to form a ‘kite’, like the one shown here.
How many lines of symmetry does the shape have?
Do you think that some shapes may have more than one line
of symmetry?
A rectangle
Take a rectangular sheet (like a post-card). Fold it once lengthwise so that one
half fits exactly over the other half. Is this fold a line of symmetry? Why?
Open it up now and
again fold on its
width in the same
way. Is this second
fold also a line of
symmetry? Why?
Do you find that these two lines are the lines of
symmetry?
A cut out from double fold
Take a rectangular piece of paper. Fold
it once and then once more. Draw
some design as shown. Cut the shape
drawn and unfold the shape. (Before
unfolding, try to guess the shape you
are likely to get).
How many lines of symmetry
does the shape have  which has been
cut out?
Create more such designs.
Do This
1st fold
 2nd fold
Form as many
shapes as you
can by
combining two
or more set
squares.  Draw
them on squared
paper and note
their lines of
symmetry.
l
2022-23
SYMMETRY
265
13.4 Figures with Multiple (more than two) Lines of Symmetry
Take a square piece of paper. Fold it into half vertically,
fold it again into half horizontally. (i.e. you have folded
it twice). Now open out the folds and again fold the
square into half (for a third time now), but this time
along a diagonal, as shown in the figure.  Again open it
and fold it into half (for the fourth time), but this time
along the other diagonal, as shown in the figure. Open
out the fold.
How many lines of symmetry does the shape have?
We can also learn to construct figures with two lines of symmetry starting
from a small part as you did in Exercise 13.1, question 4, for figures with one
line of symmetry.
1. Let us have a figure as shown alongside.
2. We want to complete it so that we get a figure
with  two lines of symmetry. Let the two lines
of symmetry be L and M.
3. W e draw the part as shown to get a figure having
line L as a line of symmetry.
3 lines of symmetry
for an equilateral triangle
2022-23