NCERT Textbook: Triangles | Mathematics (Maths) Class 10 PDF Download

 Page 1


TRIANGLES 73
6
6.1 Introduction
You are familiar with triangles and many of their properties from your earlier classes.
In Class IX, you have studied congruence of triangles in detail. Recall that two figures
are said to be congruent, if they have the same shape and the same size. In this
chapter, we shall study about those figures which have the same shape but not necessarily
the same size. Two figures having the same shape (and not necessarily the same size)
are called similar figures. In particular, we shall discuss the similarity of triangles and
apply this knowledge in giving a simple proof of Pythagoras Theorem learnt earlier.
Can you guess how heights of mountains (say Mount Everest) or distances of
some long distant objects (say moon) have been found out? Do you think these have
TRIANGLES
2024-25
Page 2


TRIANGLES 73
6
6.1 Introduction
You are familiar with triangles and many of their properties from your earlier classes.
In Class IX, you have studied congruence of triangles in detail. Recall that two figures
are said to be congruent, if they have the same shape and the same size. In this
chapter, we shall study about those figures which have the same shape but not necessarily
the same size. Two figures having the same shape (and not necessarily the same size)
are called similar figures. In particular, we shall discuss the similarity of triangles and
apply this knowledge in giving a simple proof of Pythagoras Theorem learnt earlier.
Can you guess how heights of mountains (say Mount Everest) or distances of
some long distant objects (say moon) have been found out? Do you think these have
TRIANGLES
2024-25
74 MATHEMA TICS
been measured directly with the help of a measuring tape? In fact, all these heights
and distances have been found out using the idea of indirect measurements, which is
based on the principle of similarity of figures (see Example 7, Q.15 of Exercise 6.3
and also Chapters 8 and 9 of this book).
6.2 Similar Figures
In Class IX, you have seen that all circles with the same radii are congruent, all
squares with the same side lengths are congruent and all equilateral triangles with the
same side lengths are congruent.
Now consider any two (or more)
circles [see Fig. 6.1 (i)]. Are they
congruent? Since all of them do not
have the same radius, they are not
congruent to each other. Note that
some are congruent and some are not,
but all of them have the same shape.
So they all are, what we call, similar.
Two similar figures have the same
shape but not necessarily the same
size. Therefore, all circles are similar.
What about two (or more) squares or
two (or more) equilateral triangles
[see Fig. 6.1 (ii) and (iii)]? As observed
in the case of circles, here also all
squares are similar and all equilateral
triangles are similar.
From the above, we can say
that all congruent figures are
similar but the similar figures need
not be congruent.
Can a circle and a square be
similar? Can a triangle and a square
be similar? These questions can be
answered by just looking at the
figures (see Fig. 6.1). Evidently
these figures are not similar. (Why?)
Fig. 6.1
Fig. 6.2
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Page 3


TRIANGLES 73
6
6.1 Introduction
You are familiar with triangles and many of their properties from your earlier classes.
In Class IX, you have studied congruence of triangles in detail. Recall that two figures
are said to be congruent, if they have the same shape and the same size. In this
chapter, we shall study about those figures which have the same shape but not necessarily
the same size. Two figures having the same shape (and not necessarily the same size)
are called similar figures. In particular, we shall discuss the similarity of triangles and
apply this knowledge in giving a simple proof of Pythagoras Theorem learnt earlier.
Can you guess how heights of mountains (say Mount Everest) or distances of
some long distant objects (say moon) have been found out? Do you think these have
TRIANGLES
2024-25
74 MATHEMA TICS
been measured directly with the help of a measuring tape? In fact, all these heights
and distances have been found out using the idea of indirect measurements, which is
based on the principle of similarity of figures (see Example 7, Q.15 of Exercise 6.3
and also Chapters 8 and 9 of this book).
6.2 Similar Figures
In Class IX, you have seen that all circles with the same radii are congruent, all
squares with the same side lengths are congruent and all equilateral triangles with the
same side lengths are congruent.
Now consider any two (or more)
circles [see Fig. 6.1 (i)]. Are they
congruent? Since all of them do not
have the same radius, they are not
congruent to each other. Note that
some are congruent and some are not,
but all of them have the same shape.
So they all are, what we call, similar.
Two similar figures have the same
shape but not necessarily the same
size. Therefore, all circles are similar.
What about two (or more) squares or
two (or more) equilateral triangles
[see Fig. 6.1 (ii) and (iii)]? As observed
in the case of circles, here also all
squares are similar and all equilateral
triangles are similar.
From the above, we can say
that all congruent figures are
similar but the similar figures need
not be congruent.
Can a circle and a square be
similar? Can a triangle and a square
be similar? These questions can be
answered by just looking at the
figures (see Fig. 6.1). Evidently
these figures are not similar. (Why?)
Fig. 6.1
Fig. 6.2
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TRIANGLES 75
What can you say about the two quadrilaterals ABCD and PQRS
(see Fig 6.2)?Are they similar? These  figures appear to be similar but we cannot be
certain about it.Therefore, we must have some definition of similarity of figures and
based on this definition some rules to decide whether the two given figures are similar
or not. For this, let us look at the photographs given in Fig. 6.3:
Fig. 6.3
You will at once say that they are the photographs of the same monument
(Taj Mahal) but are in different sizes. Would you say that the three photographs  are
similar? Y es,they are.
What can you say about the two photographs of the same size of the same
person one at the age of 10 years and the other at the age of 40 years? Are these
photographs similar? These photographs are of the same size but certainly they are
not of the same shape. So, they are not similar.
What does the photographer do when she prints photographs of different sizes
from the same negative? You must  have heard about the stamp size, passport size and
postcard size photographs. She generally takes a photograph on a small size film, say
of 35mm size and then enlarges it into a bigger size, say 45mm (or 55mm). Thus, if we
consider any line segment in the smaller photograph (figure), its corresponding line
segment in the bigger photograph (figure) will be
45
35
55
or
35
??
??
??
 of that of the line segment.
This really means that every line segment of the smaller photograph is enlarged
(increased) in the ratio 35:45 (or 35:55).  It can also be said that every line segment
of the bigger photograph is reduced (decreased) in the ratio 45:35 (or 55:35). Further,
if you consider inclinations (or angles) between any pair of corresponding line segments
in the two photographs of different sizes, you shall see that these inclinations(or angles)
are always equal. This is the essence of the similarity of two figures and in particular
of two polygons. We say that:
Two polygons of the same number of sides are similar, if (i) their
corresponding angles are equal and (ii) their corresponding sides are in the
same ratio (or proportion).
2024-25
Page 4


TRIANGLES 73
6
6.1 Introduction
You are familiar with triangles and many of their properties from your earlier classes.
In Class IX, you have studied congruence of triangles in detail. Recall that two figures
are said to be congruent, if they have the same shape and the same size. In this
chapter, we shall study about those figures which have the same shape but not necessarily
the same size. Two figures having the same shape (and not necessarily the same size)
are called similar figures. In particular, we shall discuss the similarity of triangles and
apply this knowledge in giving a simple proof of Pythagoras Theorem learnt earlier.
Can you guess how heights of mountains (say Mount Everest) or distances of
some long distant objects (say moon) have been found out? Do you think these have
TRIANGLES
2024-25
74 MATHEMA TICS
been measured directly with the help of a measuring tape? In fact, all these heights
and distances have been found out using the idea of indirect measurements, which is
based on the principle of similarity of figures (see Example 7, Q.15 of Exercise 6.3
and also Chapters 8 and 9 of this book).
6.2 Similar Figures
In Class IX, you have seen that all circles with the same radii are congruent, all
squares with the same side lengths are congruent and all equilateral triangles with the
same side lengths are congruent.
Now consider any two (or more)
circles [see Fig. 6.1 (i)]. Are they
congruent? Since all of them do not
have the same radius, they are not
congruent to each other. Note that
some are congruent and some are not,
but all of them have the same shape.
So they all are, what we call, similar.
Two similar figures have the same
shape but not necessarily the same
size. Therefore, all circles are similar.
What about two (or more) squares or
two (or more) equilateral triangles
[see Fig. 6.1 (ii) and (iii)]? As observed
in the case of circles, here also all
squares are similar and all equilateral
triangles are similar.
From the above, we can say
that all congruent figures are
similar but the similar figures need
not be congruent.
Can a circle and a square be
similar? Can a triangle and a square
be similar? These questions can be
answered by just looking at the
figures (see Fig. 6.1). Evidently
these figures are not similar. (Why?)
Fig. 6.1
Fig. 6.2
2024-25
TRIANGLES 75
What can you say about the two quadrilaterals ABCD and PQRS
(see Fig 6.2)?Are they similar? These  figures appear to be similar but we cannot be
certain about it.Therefore, we must have some definition of similarity of figures and
based on this definition some rules to decide whether the two given figures are similar
or not. For this, let us look at the photographs given in Fig. 6.3:
Fig. 6.3
You will at once say that they are the photographs of the same monument
(Taj Mahal) but are in different sizes. Would you say that the three photographs  are
similar? Y es,they are.
What can you say about the two photographs of the same size of the same
person one at the age of 10 years and the other at the age of 40 years? Are these
photographs similar? These photographs are of the same size but certainly they are
not of the same shape. So, they are not similar.
What does the photographer do when she prints photographs of different sizes
from the same negative? You must  have heard about the stamp size, passport size and
postcard size photographs. She generally takes a photograph on a small size film, say
of 35mm size and then enlarges it into a bigger size, say 45mm (or 55mm). Thus, if we
consider any line segment in the smaller photograph (figure), its corresponding line
segment in the bigger photograph (figure) will be
45
35
55
or
35
??
??
??
 of that of the line segment.
This really means that every line segment of the smaller photograph is enlarged
(increased) in the ratio 35:45 (or 35:55).  It can also be said that every line segment
of the bigger photograph is reduced (decreased) in the ratio 45:35 (or 55:35). Further,
if you consider inclinations (or angles) between any pair of corresponding line segments
in the two photographs of different sizes, you shall see that these inclinations(or angles)
are always equal. This is the essence of the similarity of two figures and in particular
of two polygons. We say that:
Two polygons of the same number of sides are similar, if (i) their
corresponding angles are equal and (ii) their corresponding sides are in the
same ratio (or proportion).
2024-25
76 MATHEMA TICS
Note that the same ratio of the corresponding sides is referred to as the scale
factor (or the Representative Fraction) for the polygons. You must have heard that
world maps (i.e., global maps) and blue prints for the construction of a building are
prepared using a suitable scale factor and observing certain conventions.
In order to understand similarity of figures more clearly , let us perform the following
activity:
Activity 1 :  Place a lighted bulb at a
point O on the ceiling  and directly below
it a table in your classroom. Let us cut a
polygon, say a quadrilateral ABCD, from
a plane cardboard and place this
cardboard parallel to the ground between
the lighted bulb  and the table. Then a
shadow of ABCD is cast on the table.
Mark the outline of this shadow as
A ?B ?C ?D ? (see Fig.6.4).
Note that the quadrilateral A ?B ?C ?D ?  is
an enlargement (or magnification) of  the
quadrilateral ABCD. This is because of
the property of light that light propogates
in a straight line. You may also note that
A ? lies on ray OA, B ? lies on ray OB, C ?
lies on  OC and D ? lies on OD.  Thus, quadrilaterals A ?B ?C ?D ?  and ABCD are of the
same shape but of different sizes.
So, quadrilateral  A ?B ?C ?D ?  is similiar to quadrilateral ABCD. We can also say
that quadrilateral ABCD is similar to the quadrilateral A ?B ?C ?D ?.
Here, you can also note that vertex A ? corresponds to vertex A, vertex B ?
corresponds to vertex B, vertex C ? corresponds to vertex C and vertex  D ? corresponds
to vertex D. Symbolically, these correspondences are represented as A ?? ? A, B?? ? B,
C ?? ? C and D ?? ? D. By actually measuring the angles and the sides of the two
quadrilaterals, you may verify that
(i) ? A = ? A ?, ? B = ? B ?, ? C = ? C ?, ? D = ? D ? and
(ii)  
AB BC CD DA
AB B C C D DA
?? ?
?? ?? ? ? ? ?
.
This again emphasises that two polygons of the same number of sides are
similar, if (i) all the corresponding angles are equal and (ii) all the corresponding
sides are in the same ratio (or proportion).
Fig. 6.4
2024-25
Page 5


TRIANGLES 73
6
6.1 Introduction
You are familiar with triangles and many of their properties from your earlier classes.
In Class IX, you have studied congruence of triangles in detail. Recall that two figures
are said to be congruent, if they have the same shape and the same size. In this
chapter, we shall study about those figures which have the same shape but not necessarily
the same size. Two figures having the same shape (and not necessarily the same size)
are called similar figures. In particular, we shall discuss the similarity of triangles and
apply this knowledge in giving a simple proof of Pythagoras Theorem learnt earlier.
Can you guess how heights of mountains (say Mount Everest) or distances of
some long distant objects (say moon) have been found out? Do you think these have
TRIANGLES
2024-25
74 MATHEMA TICS
been measured directly with the help of a measuring tape? In fact, all these heights
and distances have been found out using the idea of indirect measurements, which is
based on the principle of similarity of figures (see Example 7, Q.15 of Exercise 6.3
and also Chapters 8 and 9 of this book).
6.2 Similar Figures
In Class IX, you have seen that all circles with the same radii are congruent, all
squares with the same side lengths are congruent and all equilateral triangles with the
same side lengths are congruent.
Now consider any two (or more)
circles [see Fig. 6.1 (i)]. Are they
congruent? Since all of them do not
have the same radius, they are not
congruent to each other. Note that
some are congruent and some are not,
but all of them have the same shape.
So they all are, what we call, similar.
Two similar figures have the same
shape but not necessarily the same
size. Therefore, all circles are similar.
What about two (or more) squares or
two (or more) equilateral triangles
[see Fig. 6.1 (ii) and (iii)]? As observed
in the case of circles, here also all
squares are similar and all equilateral
triangles are similar.
From the above, we can say
that all congruent figures are
similar but the similar figures need
not be congruent.
Can a circle and a square be
similar? Can a triangle and a square
be similar? These questions can be
answered by just looking at the
figures (see Fig. 6.1). Evidently
these figures are not similar. (Why?)
Fig. 6.1
Fig. 6.2
2024-25
TRIANGLES 75
What can you say about the two quadrilaterals ABCD and PQRS
(see Fig 6.2)?Are they similar? These  figures appear to be similar but we cannot be
certain about it.Therefore, we must have some definition of similarity of figures and
based on this definition some rules to decide whether the two given figures are similar
or not. For this, let us look at the photographs given in Fig. 6.3:
Fig. 6.3
You will at once say that they are the photographs of the same monument
(Taj Mahal) but are in different sizes. Would you say that the three photographs  are
similar? Y es,they are.
What can you say about the two photographs of the same size of the same
person one at the age of 10 years and the other at the age of 40 years? Are these
photographs similar? These photographs are of the same size but certainly they are
not of the same shape. So, they are not similar.
What does the photographer do when she prints photographs of different sizes
from the same negative? You must  have heard about the stamp size, passport size and
postcard size photographs. She generally takes a photograph on a small size film, say
of 35mm size and then enlarges it into a bigger size, say 45mm (or 55mm). Thus, if we
consider any line segment in the smaller photograph (figure), its corresponding line
segment in the bigger photograph (figure) will be
45
35
55
or
35
??
??
??
 of that of the line segment.
This really means that every line segment of the smaller photograph is enlarged
(increased) in the ratio 35:45 (or 35:55).  It can also be said that every line segment
of the bigger photograph is reduced (decreased) in the ratio 45:35 (or 55:35). Further,
if you consider inclinations (or angles) between any pair of corresponding line segments
in the two photographs of different sizes, you shall see that these inclinations(or angles)
are always equal. This is the essence of the similarity of two figures and in particular
of two polygons. We say that:
Two polygons of the same number of sides are similar, if (i) their
corresponding angles are equal and (ii) their corresponding sides are in the
same ratio (or proportion).
2024-25
76 MATHEMA TICS
Note that the same ratio of the corresponding sides is referred to as the scale
factor (or the Representative Fraction) for the polygons. You must have heard that
world maps (i.e., global maps) and blue prints for the construction of a building are
prepared using a suitable scale factor and observing certain conventions.
In order to understand similarity of figures more clearly , let us perform the following
activity:
Activity 1 :  Place a lighted bulb at a
point O on the ceiling  and directly below
it a table in your classroom. Let us cut a
polygon, say a quadrilateral ABCD, from
a plane cardboard and place this
cardboard parallel to the ground between
the lighted bulb  and the table. Then a
shadow of ABCD is cast on the table.
Mark the outline of this shadow as
A ?B ?C ?D ? (see Fig.6.4).
Note that the quadrilateral A ?B ?C ?D ?  is
an enlargement (or magnification) of  the
quadrilateral ABCD. This is because of
the property of light that light propogates
in a straight line. You may also note that
A ? lies on ray OA, B ? lies on ray OB, C ?
lies on  OC and D ? lies on OD.  Thus, quadrilaterals A ?B ?C ?D ?  and ABCD are of the
same shape but of different sizes.
So, quadrilateral  A ?B ?C ?D ?  is similiar to quadrilateral ABCD. We can also say
that quadrilateral ABCD is similar to the quadrilateral A ?B ?C ?D ?.
Here, you can also note that vertex A ? corresponds to vertex A, vertex B ?
corresponds to vertex B, vertex C ? corresponds to vertex C and vertex  D ? corresponds
to vertex D. Symbolically, these correspondences are represented as A ?? ? A, B?? ? B,
C ?? ? C and D ?? ? D. By actually measuring the angles and the sides of the two
quadrilaterals, you may verify that
(i) ? A = ? A ?, ? B = ? B ?, ? C = ? C ?, ? D = ? D ? and
(ii)  
AB BC CD DA
AB B C C D DA
?? ?
?? ?? ? ? ? ?
.
This again emphasises that two polygons of the same number of sides are
similar, if (i) all the corresponding angles are equal and (ii) all the corresponding
sides are in the same ratio (or proportion).
Fig. 6.4
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TRIANGLES 77
From the above, you can easily say that quadrilaterals ABCD and PQRS of
Fig. 6.5 are similar.
Fig. 6.5
Remark : You can verify that if one polygon is similar to another polygon and this
second polygon is similar to a third polygon, then the first polygon is similar to the third
polygon.
You may note that in the two quadrilaterals (a square and a rectangle) of
Fig. 6.6, corresponding angles are equal, but their corresponding sides are not in the
same ratio.
Fig. 6.6
So, the two quadrilaterals are not similar. Similarly, you may note that in the two
quadrilaterals (a square and a rhombus) of Fig. 6.7, corresponding sides are in the
same ratio, but their corresponding angles are not equal. Again, the two polygons
(quadrilaterals) are not similar.
2024-25