Class 7 Maths Chapter 7 NCERT Book - A Tale of Three Intersecting Lines

 Page 1


A TALE OF THREE 
INTERSECTING 
LINES
7
A triangle is the most basic closed shape. As we know, it consists of:
•  three corner points, that we call the vertex of the triangle, and 
•  three line segments or the sides of the triangle that join the pairs 
of vertices.
Triangles come in various shapes. Some of them are shown below.
B
C
? ABC
A
X
Z Y
? YZX
?WUV
V
U
W
?BAU
A
B
U
Observe the symbol used to denote a triangle and how the triangles 
are named using their vertices. While naming a triangle, the vertices 
can come in any order.
The three sides meeting at the corners give rise to three angles that 
we call the angles of the triangle. For example, in ?ABC, these angles 
are ?CAB, ?ABC, ?BCA, which we simply denote as ?A, ?B and ?C, 
respectively.
What happens when the three vertices lie on a straight line?
7.1 Equilateral Triangles
Among all the triangles, the equilateral triangles are the most symmetric 
ones. These are triangles in which all the sides are of equal lengths. Let 
us try constructing them.
Construct a triangle in which all the sides are of length 4 cm.
Chapter-7.indd   146 Chapter-7.indd   146 4/11/2025   7:31:55 PM 4/11/2025   7:31:55 PM
Page 2


A TALE OF THREE 
INTERSECTING 
LINES
7
A triangle is the most basic closed shape. As we know, it consists of:
•  three corner points, that we call the vertex of the triangle, and 
•  three line segments or the sides of the triangle that join the pairs 
of vertices.
Triangles come in various shapes. Some of them are shown below.
B
C
? ABC
A
X
Z Y
? YZX
?WUV
V
U
W
?BAU
A
B
U
Observe the symbol used to denote a triangle and how the triangles 
are named using their vertices. While naming a triangle, the vertices 
can come in any order.
The three sides meeting at the corners give rise to three angles that 
we call the angles of the triangle. For example, in ?ABC, these angles 
are ?CAB, ?ABC, ?BCA, which we simply denote as ?A, ?B and ?C, 
respectively.
What happens when the three vertices lie on a straight line?
7.1 Equilateral Triangles
Among all the triangles, the equilateral triangles are the most symmetric 
ones. These are triangles in which all the sides are of equal lengths. Let 
us try constructing them.
Construct a triangle in which all the sides are of length 4 cm.
Chapter-7.indd   146 Chapter-7.indd   146 4/11/2025   7:31:55 PM 4/11/2025   7:31:55 PM
A Tale of Three Intersecting Lines
How did you construct this triangle and what tools did you use? Can 
this construction be done only using a marked ruler (and a pencil)?
Constructing this triangle using just a ruler is certainly possible. But 
this might require several trials. Say we draw the base?—?let us call it 
AB?—?of length 4 cm (see the figure below), and mark the third point C 
using a ruler such that AC = 4 cm. This may not lead to BC also having 
a length of 4 cm. If this happens, we will have to keep making attempts 
to mark C till we get BC to be 4 cm long.
A B
C
4 cm?
4 cm
4 cm
How do we make this construction more efficient?
Recall solving a similar problem in the previous year using a compass 
(in the Chapter ‘Playing with Constructions’). We had to mark the top 
point of a ‘house’ which is 5 cm from two other points. The method we 
used to get that point can also be used here.
After constructing AB = 4 cm, we can do the following.
Step 1: Using a compass, construct a sufficiently long arc of radius 4 cm 
from A, as shown in the figure. The point C is somewhere on this arc. 
How do we mark it?
A
B
4 cm
147
Chapter-7.indd   147 Chapter-7.indd   147 4/11/2025   7:31:55 PM 4/11/2025   7:31:55 PM
Page 3


A TALE OF THREE 
INTERSECTING 
LINES
7
A triangle is the most basic closed shape. As we know, it consists of:
•  three corner points, that we call the vertex of the triangle, and 
•  three line segments or the sides of the triangle that join the pairs 
of vertices.
Triangles come in various shapes. Some of them are shown below.
B
C
? ABC
A
X
Z Y
? YZX
?WUV
V
U
W
?BAU
A
B
U
Observe the symbol used to denote a triangle and how the triangles 
are named using their vertices. While naming a triangle, the vertices 
can come in any order.
The three sides meeting at the corners give rise to three angles that 
we call the angles of the triangle. For example, in ?ABC, these angles 
are ?CAB, ?ABC, ?BCA, which we simply denote as ?A, ?B and ?C, 
respectively.
What happens when the three vertices lie on a straight line?
7.1 Equilateral Triangles
Among all the triangles, the equilateral triangles are the most symmetric 
ones. These are triangles in which all the sides are of equal lengths. Let 
us try constructing them.
Construct a triangle in which all the sides are of length 4 cm.
Chapter-7.indd   146 Chapter-7.indd   146 4/11/2025   7:31:55 PM 4/11/2025   7:31:55 PM
A Tale of Three Intersecting Lines
How did you construct this triangle and what tools did you use? Can 
this construction be done only using a marked ruler (and a pencil)?
Constructing this triangle using just a ruler is certainly possible. But 
this might require several trials. Say we draw the base?—?let us call it 
AB?—?of length 4 cm (see the figure below), and mark the third point C 
using a ruler such that AC = 4 cm. This may not lead to BC also having 
a length of 4 cm. If this happens, we will have to keep making attempts 
to mark C till we get BC to be 4 cm long.
A B
C
4 cm?
4 cm
4 cm
How do we make this construction more efficient?
Recall solving a similar problem in the previous year using a compass 
(in the Chapter ‘Playing with Constructions’). We had to mark the top 
point of a ‘house’ which is 5 cm from two other points. The method we 
used to get that point can also be used here.
After constructing AB = 4 cm, we can do the following.
Step 1: Using a compass, construct a sufficiently long arc of radius 4 cm 
from A, as shown in the figure. The point C is somewhere on this arc. 
How do we mark it?
A
B
4 cm
147
Chapter-7.indd   147 Chapter-7.indd   147 4/11/2025   7:31:55 PM 4/11/2025   7:31:55 PM
Ganita Prakash | Grade 7
Step 2: Construct another arc of radius 4 cm from B.
A
B
4 cm
Let C be the point of intersection of the arcs.
The construction ensures that both AC and BC are of length 4 cm. Can 
you see why?
Step 3: Join AC and BC to get the required equilateral triangle.
A
B
4 cm
C
7.2  Constructing a Triangle When its Sides are 
Given
How do we construct triangles that are not equilateral?
Construct a triangle of sidelength 4 cm, 5 cm and 6 cm.
As in the previous case, this triangle can also be constructed using 
just a marked ruler. But it will involve several trials.
148
Chapter-7.indd   148 Chapter-7.indd   148 4/11/2025   7:31:55 PM 4/11/2025   7:31:55 PM
Page 4


A TALE OF THREE 
INTERSECTING 
LINES
7
A triangle is the most basic closed shape. As we know, it consists of:
•  three corner points, that we call the vertex of the triangle, and 
•  three line segments or the sides of the triangle that join the pairs 
of vertices.
Triangles come in various shapes. Some of them are shown below.
B
C
? ABC
A
X
Z Y
? YZX
?WUV
V
U
W
?BAU
A
B
U
Observe the symbol used to denote a triangle and how the triangles 
are named using their vertices. While naming a triangle, the vertices 
can come in any order.
The three sides meeting at the corners give rise to three angles that 
we call the angles of the triangle. For example, in ?ABC, these angles 
are ?CAB, ?ABC, ?BCA, which we simply denote as ?A, ?B and ?C, 
respectively.
What happens when the three vertices lie on a straight line?
7.1 Equilateral Triangles
Among all the triangles, the equilateral triangles are the most symmetric 
ones. These are triangles in which all the sides are of equal lengths. Let 
us try constructing them.
Construct a triangle in which all the sides are of length 4 cm.
Chapter-7.indd   146 Chapter-7.indd   146 4/11/2025   7:31:55 PM 4/11/2025   7:31:55 PM
A Tale of Three Intersecting Lines
How did you construct this triangle and what tools did you use? Can 
this construction be done only using a marked ruler (and a pencil)?
Constructing this triangle using just a ruler is certainly possible. But 
this might require several trials. Say we draw the base?—?let us call it 
AB?—?of length 4 cm (see the figure below), and mark the third point C 
using a ruler such that AC = 4 cm. This may not lead to BC also having 
a length of 4 cm. If this happens, we will have to keep making attempts 
to mark C till we get BC to be 4 cm long.
A B
C
4 cm?
4 cm
4 cm
How do we make this construction more efficient?
Recall solving a similar problem in the previous year using a compass 
(in the Chapter ‘Playing with Constructions’). We had to mark the top 
point of a ‘house’ which is 5 cm from two other points. The method we 
used to get that point can also be used here.
After constructing AB = 4 cm, we can do the following.
Step 1: Using a compass, construct a sufficiently long arc of radius 4 cm 
from A, as shown in the figure. The point C is somewhere on this arc. 
How do we mark it?
A
B
4 cm
147
Chapter-7.indd   147 Chapter-7.indd   147 4/11/2025   7:31:55 PM 4/11/2025   7:31:55 PM
Ganita Prakash | Grade 7
Step 2: Construct another arc of radius 4 cm from B.
A
B
4 cm
Let C be the point of intersection of the arcs.
The construction ensures that both AC and BC are of length 4 cm. Can 
you see why?
Step 3: Join AC and BC to get the required equilateral triangle.
A
B
4 cm
C
7.2  Constructing a Triangle When its Sides are 
Given
How do we construct triangles that are not equilateral?
Construct a triangle of sidelength 4 cm, 5 cm and 6 cm.
As in the previous case, this triangle can also be constructed using 
just a marked ruler. But it will involve several trials.
148
Chapter-7.indd   148 Chapter-7.indd   148 4/11/2025   7:31:55 PM 4/11/2025   7:31:55 PM
A Tale of Three Intersecting Lines
How do we construct this triangle more efficiently?
Choose one of the given lengths to be the base of the triangle: say 4 
cm. Draw the base. Let A and B be the base vertices, and call the third 
vertex C. Let AC = 5 cm and BC = 6 cm.
Fig. 7.1
Like we did in the case of equilateral triangles, let us first get all the 
points that are at a 5 cm distance from A. These points lie on the circle 
whose centre is A and has radius 5 cm. The point C must lie somewhere 
on this circle. How do we find it?
4 cm
A B
5 cm
Fig. 7.2
We will make use of the fact that the point C is 6 cm away from B.
Construct an arc of radius 6 cm from B.
A B
5 cm
6 cm
Fig. 7.3
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Page 5


A TALE OF THREE 
INTERSECTING 
LINES
7
A triangle is the most basic closed shape. As we know, it consists of:
•  three corner points, that we call the vertex of the triangle, and 
•  three line segments or the sides of the triangle that join the pairs 
of vertices.
Triangles come in various shapes. Some of them are shown below.
B
C
? ABC
A
X
Z Y
? YZX
?WUV
V
U
W
?BAU
A
B
U
Observe the symbol used to denote a triangle and how the triangles 
are named using their vertices. While naming a triangle, the vertices 
can come in any order.
The three sides meeting at the corners give rise to three angles that 
we call the angles of the triangle. For example, in ?ABC, these angles 
are ?CAB, ?ABC, ?BCA, which we simply denote as ?A, ?B and ?C, 
respectively.
What happens when the three vertices lie on a straight line?
7.1 Equilateral Triangles
Among all the triangles, the equilateral triangles are the most symmetric 
ones. These are triangles in which all the sides are of equal lengths. Let 
us try constructing them.
Construct a triangle in which all the sides are of length 4 cm.
Chapter-7.indd   146 Chapter-7.indd   146 4/11/2025   7:31:55 PM 4/11/2025   7:31:55 PM
A Tale of Three Intersecting Lines
How did you construct this triangle and what tools did you use? Can 
this construction be done only using a marked ruler (and a pencil)?
Constructing this triangle using just a ruler is certainly possible. But 
this might require several trials. Say we draw the base?—?let us call it 
AB?—?of length 4 cm (see the figure below), and mark the third point C 
using a ruler such that AC = 4 cm. This may not lead to BC also having 
a length of 4 cm. If this happens, we will have to keep making attempts 
to mark C till we get BC to be 4 cm long.
A B
C
4 cm?
4 cm
4 cm
How do we make this construction more efficient?
Recall solving a similar problem in the previous year using a compass 
(in the Chapter ‘Playing with Constructions’). We had to mark the top 
point of a ‘house’ which is 5 cm from two other points. The method we 
used to get that point can also be used here.
After constructing AB = 4 cm, we can do the following.
Step 1: Using a compass, construct a sufficiently long arc of radius 4 cm 
from A, as shown in the figure. The point C is somewhere on this arc. 
How do we mark it?
A
B
4 cm
147
Chapter-7.indd   147 Chapter-7.indd   147 4/11/2025   7:31:55 PM 4/11/2025   7:31:55 PM
Ganita Prakash | Grade 7
Step 2: Construct another arc of radius 4 cm from B.
A
B
4 cm
Let C be the point of intersection of the arcs.
The construction ensures that both AC and BC are of length 4 cm. Can 
you see why?
Step 3: Join AC and BC to get the required equilateral triangle.
A
B
4 cm
C
7.2  Constructing a Triangle When its Sides are 
Given
How do we construct triangles that are not equilateral?
Construct a triangle of sidelength 4 cm, 5 cm and 6 cm.
As in the previous case, this triangle can also be constructed using 
just a marked ruler. But it will involve several trials.
148
Chapter-7.indd   148 Chapter-7.indd   148 4/11/2025   7:31:55 PM 4/11/2025   7:31:55 PM
A Tale of Three Intersecting Lines
How do we construct this triangle more efficiently?
Choose one of the given lengths to be the base of the triangle: say 4 
cm. Draw the base. Let A and B be the base vertices, and call the third 
vertex C. Let AC = 5 cm and BC = 6 cm.
Fig. 7.1
Like we did in the case of equilateral triangles, let us first get all the 
points that are at a 5 cm distance from A. These points lie on the circle 
whose centre is A and has radius 5 cm. The point C must lie somewhere 
on this circle. How do we find it?
4 cm
A B
5 cm
Fig. 7.2
We will make use of the fact that the point C is 6 cm away from B.
Construct an arc of radius 6 cm from B.
A B
5 cm
6 cm
Fig. 7.3
149
Chapter-7.indd   149 Chapter-7.indd   149 4/11/2025   7:31:55 PM 4/11/2025   7:31:55 PM
Ganita Prakash | Grade 7
The required point C is one of the points of intersection of the two 
circles.
The reason why the point of 
intersection is the third vertex is the 
same as for equilateral triangles. This 
point lies on both the circles. Hence 
its distance from A is the radius of 
the circle centred at A (5 cm) and its 
distance from B is the same as the 
radius of the circle centred at B (6 cm).
Let us summarise the steps of 
construction, noting that constructing 
full circles is not necessary to get the 
third vertex (see Fig. 7.2 and 7.3).
Step 1:  Construct the base AB with one of the side lengths. Let us choose 
AB = 4 cm (see Fig. 7.1).
Step 2:  From A, construct a sufficiently long arc of radius 5 cm  
(see Fig. 7.2).
Step 3:  From B, construct an arc of radius 6 cm such that it intersects 
the first arc (see Fig. 7.3).
Step 4:  The point where both the arcs meet is the required third 
vertex C. Join AC and BC to get ?ABC.
Construct
Construct triangles having the following sidelengths (all the units are 
in cm):
(a)  4, 4, 6
(b)  3, 4, 5
(c)  1, 5, 5
(d)  4, 6, 8
(e)  3.5, 3.5, 3.5
We have seen that triangles having all three equal sides are called 
equilateral triangles. Those having two equal sides are called isosceles 
triangles.
Figure it Out
1.  Use the points on the circle and/or the centre to 
form isosceles triangles.
A B
C
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