NCERT Textbook: Symmetry

# 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.
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.
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
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.
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
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.
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
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.
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
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
```

## NCERT Textbooks (Class 6 to Class 12)

681 docs|672 tests

## FAQs on NCERT Textbook: Symmetry - NCERT Textbooks (Class 6 to Class 12) - CTET & State TET

 1. What is symmetry?
Ans. Symmetry is a concept in mathematics that refers to a balanced arrangement of elements or objects. It is the property of an object being unchanged when certain transformations, such as reflection, rotation, or translation, are applied to it. In simpler terms, symmetry means that one half of an object is a mirror image of the other half.
 2. How is symmetry represented?
Ans. Symmetry can be represented using various notations and symbols. In geometry, we often use the letter 'S' to denote symmetry. A line of symmetry is represented by a dashed line that divides an object into two equal halves, with each half being a mirror image of the other. For rotational symmetry, we use the notation 'n-fold symmetry' to indicate the number of times an object can be rotated to coincide with its original position.
 3. What are the different types of symmetry?
Ans. There are several types of symmetry, including: - Reflectional Symmetry: Also known as line symmetry, it occurs when an object can be divided into two equal halves by a straight line. The two halves are mirror images of each other. - Rotational Symmetry: This type of symmetry occurs when an object can be rotated around a central point and still maintain its original appearance. The number of times an object can be rotated to coincide with its original position represents its rotational symmetry. - Translational Symmetry: It occurs when an object can be shifted or translated along a straight line without changing its overall appearance. - Point Symmetry: Point symmetry is present when an object remains unchanged after a 180-degree rotation about a single point.
 4. How is symmetry useful in real-life applications?
Ans. Symmetry has various applications in our daily lives and different fields. Some examples include: - Architecture: Symmetry is often used in building designs to create visually pleasing and balanced structures. Many famous landmarks and buildings, such as the Taj Mahal, exhibit symmetrical patterns. - Art and Design: Artists and designers often utilize symmetry to create aesthetically pleasing compositions. Symmetrical patterns are commonly found in paintings, sculptures, and even product designs. - Science and Nature: Symmetry plays a crucial role in understanding and describing patterns and structures in the natural world. From the arrangement of petals in a flower to the molecular symmetry in crystals, symmetry is present in various scientific phenomena. - Engineering: Engineers often rely on symmetry to design and build structurally sound and balanced structures. Symmetrical designs help distribute forces and loads evenly, enhancing stability and performance.
 5. How can symmetry be identified in shapes and objects?
Ans. Symmetry can be identified by observing certain characteristics in shapes and objects. Here are some common methods: - Reflectional Symmetry: Look for objects or shapes that can be divided into two equal halves by a straight line. If the two halves are mirror images of each other, the object exhibits reflectional symmetry. - Rotational Symmetry: Rotate the object around a central point and observe if it maintains its original appearance at certain intervals. The number of times it aligns with the original position represents its rotational symmetry. - Translational Symmetry: Look for shapes or objects that can be shifted or translated along a straight line without changing their overall appearance. If the shifted position aligns with the original position, translational symmetry is present. - Point Symmetry: Determine if an object remains unchanged after a 180-degree rotation about a single point. If it does, it exhibits point symmetry.

## NCERT Textbooks (Class 6 to Class 12)

681 docs|672 tests

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