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

 Page 1


PRACTICAL GEOMETRY  57
DO THIS
4.1  Introduction
Y ou have learnt how to draw triangles in Class VII. We require three measurements
(of sides and angles) to draw a unique triangle.
Since three measurements were enough to draw a triangle, a natural question arises
whether four measurements would be sufficient to draw a unique four sided closed figure,
namely , a quadrilateral.
T ake a pair of sticks of equal lengths, say
10 cm. Take another pair of sticks of
equal lengths, say, 8 cm. Hinge them up
suitably to get a rectangle of length 10 cm
and breadth 8 cm.
This rectangle has been created with
the 4 available measurements.
Now just push along the breadth of
the rectangle. Is the new shape obtained,
still a rectangle (Fig 4.2)? Observe
that the rectangle has now become
a parallelogram. Have you altered the
lengths of the sticks? No! The
measurements of sides remain the same.
Give another push to the newly
obtained shape in a different direction;
what do you get? You again get a
parallelogram, which is altogether different
(Fig 4.3),  yet the four measurements
remain the same.
This shows that 4 measurements of a quadrilateral cannot determine it uniquely .
Can 5 measurements determine a quadrilateral uniquely? Let us go back to the activity!
Practical Geometry
CHAPTER
4
Fig 4.1
Fig 4.2
Fig 4.3
2022-23
Page 2


PRACTICAL GEOMETRY  57
DO THIS
4.1  Introduction
Y ou have learnt how to draw triangles in Class VII. We require three measurements
(of sides and angles) to draw a unique triangle.
Since three measurements were enough to draw a triangle, a natural question arises
whether four measurements would be sufficient to draw a unique four sided closed figure,
namely , a quadrilateral.
T ake a pair of sticks of equal lengths, say
10 cm. Take another pair of sticks of
equal lengths, say, 8 cm. Hinge them up
suitably to get a rectangle of length 10 cm
and breadth 8 cm.
This rectangle has been created with
the 4 available measurements.
Now just push along the breadth of
the rectangle. Is the new shape obtained,
still a rectangle (Fig 4.2)? Observe
that the rectangle has now become
a parallelogram. Have you altered the
lengths of the sticks? No! The
measurements of sides remain the same.
Give another push to the newly
obtained shape in a different direction;
what do you get? You again get a
parallelogram, which is altogether different
(Fig 4.3),  yet the four measurements
remain the same.
This shows that 4 measurements of a quadrilateral cannot determine it uniquely .
Can 5 measurements determine a quadrilateral uniquely? Let us go back to the activity!
Practical Geometry
CHAPTER
4
Fig 4.1
Fig 4.2
Fig 4.3
2022-23
58  MATHEMATICS
THINK, DISCUSS AND WRITE
Y ou have constructed a rectangle with
two sticks each of length 10 cm and other
two sticks each of length 8 cm. Now
introduce another stick of length equal to
BD and tie it along BD (Fig 4.4). If you
push the breadth now, does the shape
change? No! It cannot, without making the
figure open. The introduction of the fifth
stick has fixed the rectangle uniquely, i.e.,
there is no other quadrilateral (with the
given lengths of sides) possible now .
Thus, we observe that five measurements can determine a quadrilateral uniquely .
But will any five measurements (of sides and angles) be sufficient to draw a unique
quadrilateral?
Arshad has five measurements of a quadrilateral ABCD. These are AB = 5 cm,
?A = 50°, AC = 4 cm, BD = 5 cm and AD = 6 cm. Can he construct a unique
quadrilateral? Give reasons for your answer.
4.2 Constructing a Quadrilateral
We shall learn how to construct a unique quadrilateral given the following
measurements:
• When four sides and one diagonal are given.
• When two diagonals and three sides are given.
• When two adjacent sides and three angles are given.
• When three sides and two included angles are given.
• When other special properties are known.
Let us take up these constructions one-by-one.
4.2.1 When the lengths of four  sides and a diagonal are given
W e shall explain this construction through an example.
Example 1: Construct a quadrilateral PQRS
where PQ = 4 cm,QR = 6 cm, RS = 5 cm,
PS = 5.5 cm and PR = 7 cm.
Solution: [A rough sketch will help us in
visualising the quadrilateral. W e draw this first and
mark the measurements.] (Fig 4.5)
Fig 4.4
Fig 4.5
2022-23
Page 3


PRACTICAL GEOMETRY  57
DO THIS
4.1  Introduction
Y ou have learnt how to draw triangles in Class VII. We require three measurements
(of sides and angles) to draw a unique triangle.
Since three measurements were enough to draw a triangle, a natural question arises
whether four measurements would be sufficient to draw a unique four sided closed figure,
namely , a quadrilateral.
T ake a pair of sticks of equal lengths, say
10 cm. Take another pair of sticks of
equal lengths, say, 8 cm. Hinge them up
suitably to get a rectangle of length 10 cm
and breadth 8 cm.
This rectangle has been created with
the 4 available measurements.
Now just push along the breadth of
the rectangle. Is the new shape obtained,
still a rectangle (Fig 4.2)? Observe
that the rectangle has now become
a parallelogram. Have you altered the
lengths of the sticks? No! The
measurements of sides remain the same.
Give another push to the newly
obtained shape in a different direction;
what do you get? You again get a
parallelogram, which is altogether different
(Fig 4.3),  yet the four measurements
remain the same.
This shows that 4 measurements of a quadrilateral cannot determine it uniquely .
Can 5 measurements determine a quadrilateral uniquely? Let us go back to the activity!
Practical Geometry
CHAPTER
4
Fig 4.1
Fig 4.2
Fig 4.3
2022-23
58  MATHEMATICS
THINK, DISCUSS AND WRITE
Y ou have constructed a rectangle with
two sticks each of length 10 cm and other
two sticks each of length 8 cm. Now
introduce another stick of length equal to
BD and tie it along BD (Fig 4.4). If you
push the breadth now, does the shape
change? No! It cannot, without making the
figure open. The introduction of the fifth
stick has fixed the rectangle uniquely, i.e.,
there is no other quadrilateral (with the
given lengths of sides) possible now .
Thus, we observe that five measurements can determine a quadrilateral uniquely .
But will any five measurements (of sides and angles) be sufficient to draw a unique
quadrilateral?
Arshad has five measurements of a quadrilateral ABCD. These are AB = 5 cm,
?A = 50°, AC = 4 cm, BD = 5 cm and AD = 6 cm. Can he construct a unique
quadrilateral? Give reasons for your answer.
4.2 Constructing a Quadrilateral
We shall learn how to construct a unique quadrilateral given the following
measurements:
• When four sides and one diagonal are given.
• When two diagonals and three sides are given.
• When two adjacent sides and three angles are given.
• When three sides and two included angles are given.
• When other special properties are known.
Let us take up these constructions one-by-one.
4.2.1 When the lengths of four  sides and a diagonal are given
W e shall explain this construction through an example.
Example 1: Construct a quadrilateral PQRS
where PQ = 4 cm,QR = 6 cm, RS = 5 cm,
PS = 5.5 cm and PR = 7 cm.
Solution: [A rough sketch will help us in
visualising the quadrilateral. W e draw this first and
mark the measurements.] (Fig 4.5)
Fig 4.4
Fig 4.5
2022-23
PRACTICAL GEOMETRY  59
Step 1 From the rough sketch, it is easy to see that ?PQR
can be constructed using SSS construction condition.
Draw ?PQR (Fig 4.6).
Step 2 Now, we have to locate the fourth point S. This ‘S’
would be on the side opposite to Q with reference to
PR. For that, we have two measurements.
S is 5.5 cm away from P . So, with P as centre, draw
an arc of radius 5.5 cm. (The point S is somewhere
on this arc!) (Fig 4.7).
Step 3 S is 5 cm away from R. So with R as centre, draw an arc of radius 5 cm (The
point S is somewhere on this arc also!) (Fig 4.8).
Fig 4.6
Fig 4.7
Fig 4.8
2022-23
Page 4


PRACTICAL GEOMETRY  57
DO THIS
4.1  Introduction
Y ou have learnt how to draw triangles in Class VII. We require three measurements
(of sides and angles) to draw a unique triangle.
Since three measurements were enough to draw a triangle, a natural question arises
whether four measurements would be sufficient to draw a unique four sided closed figure,
namely , a quadrilateral.
T ake a pair of sticks of equal lengths, say
10 cm. Take another pair of sticks of
equal lengths, say, 8 cm. Hinge them up
suitably to get a rectangle of length 10 cm
and breadth 8 cm.
This rectangle has been created with
the 4 available measurements.
Now just push along the breadth of
the rectangle. Is the new shape obtained,
still a rectangle (Fig 4.2)? Observe
that the rectangle has now become
a parallelogram. Have you altered the
lengths of the sticks? No! The
measurements of sides remain the same.
Give another push to the newly
obtained shape in a different direction;
what do you get? You again get a
parallelogram, which is altogether different
(Fig 4.3),  yet the four measurements
remain the same.
This shows that 4 measurements of a quadrilateral cannot determine it uniquely .
Can 5 measurements determine a quadrilateral uniquely? Let us go back to the activity!
Practical Geometry
CHAPTER
4
Fig 4.1
Fig 4.2
Fig 4.3
2022-23
58  MATHEMATICS
THINK, DISCUSS AND WRITE
Y ou have constructed a rectangle with
two sticks each of length 10 cm and other
two sticks each of length 8 cm. Now
introduce another stick of length equal to
BD and tie it along BD (Fig 4.4). If you
push the breadth now, does the shape
change? No! It cannot, without making the
figure open. The introduction of the fifth
stick has fixed the rectangle uniquely, i.e.,
there is no other quadrilateral (with the
given lengths of sides) possible now .
Thus, we observe that five measurements can determine a quadrilateral uniquely .
But will any five measurements (of sides and angles) be sufficient to draw a unique
quadrilateral?
Arshad has five measurements of a quadrilateral ABCD. These are AB = 5 cm,
?A = 50°, AC = 4 cm, BD = 5 cm and AD = 6 cm. Can he construct a unique
quadrilateral? Give reasons for your answer.
4.2 Constructing a Quadrilateral
We shall learn how to construct a unique quadrilateral given the following
measurements:
• When four sides and one diagonal are given.
• When two diagonals and three sides are given.
• When two adjacent sides and three angles are given.
• When three sides and two included angles are given.
• When other special properties are known.
Let us take up these constructions one-by-one.
4.2.1 When the lengths of four  sides and a diagonal are given
W e shall explain this construction through an example.
Example 1: Construct a quadrilateral PQRS
where PQ = 4 cm,QR = 6 cm, RS = 5 cm,
PS = 5.5 cm and PR = 7 cm.
Solution: [A rough sketch will help us in
visualising the quadrilateral. W e draw this first and
mark the measurements.] (Fig 4.5)
Fig 4.4
Fig 4.5
2022-23
PRACTICAL GEOMETRY  59
Step 1 From the rough sketch, it is easy to see that ?PQR
can be constructed using SSS construction condition.
Draw ?PQR (Fig 4.6).
Step 2 Now, we have to locate the fourth point S. This ‘S’
would be on the side opposite to Q with reference to
PR. For that, we have two measurements.
S is 5.5 cm away from P . So, with P as centre, draw
an arc of radius 5.5 cm. (The point S is somewhere
on this arc!) (Fig 4.7).
Step 3 S is 5 cm away from R. So with R as centre, draw an arc of radius 5 cm (The
point S is somewhere on this arc also!) (Fig 4.8).
Fig 4.6
Fig 4.7
Fig 4.8
2022-23
60  MATHEMATICS
THINK, DISCUSS AND WRITE
Step 4 S should lie on both the arcs drawn.
So it is the point of intersection of the
two arcs. Mark S and complete PQRS.
PQRS is the required quadrilateral
(Fig 4.9).
(i) We saw that 5 measurements of a quadrilateral can determine a quadrilateral
uniquely. Do you think any five measurements of the quadrilateral can do this?
(ii) Can you draw a parallelogram BATS where BA = 5 cm, AT = 6 cm and
AS = 6.5 cm? Why?
(iii) Can you draw a rhombus ZEAL where ZE = 3.5 cm, diagonal EL = 5 cm? Why?
(iv) A student attempted to draw a quadrilateral PLA Y where PL = 3 cm, LA = 4 cm,
AY = 4.5 cm, PY = 2 cm and LY = 6 cm, but could not draw it. What is
the reason?
[Hint: Discuss it using a rough sketch].
EXERCISE 4.1
1. Construct the following quadrilaterals.
(i) Quadrilateral ABCD. (ii) Quadrilateral JUMP
AB = 4.5 cm JU = 3.5 cm
BC = 5.5 cm UM = 4 cm
CD = 4 cm MP = 5 cm
AD = 6 cm PJ = 4.5 cm
AC = 7 cm PU = 6.5 cm
(iii) Parallelogram MORE (iv) Rhombus BEST
OR = 6 cm BE = 4.5 cm
RE = 4.5 cm ET = 6 cm
EO = 7.5 cm
Fig 4.9
2022-23
Page 5


PRACTICAL GEOMETRY  57
DO THIS
4.1  Introduction
Y ou have learnt how to draw triangles in Class VII. We require three measurements
(of sides and angles) to draw a unique triangle.
Since three measurements were enough to draw a triangle, a natural question arises
whether four measurements would be sufficient to draw a unique four sided closed figure,
namely , a quadrilateral.
T ake a pair of sticks of equal lengths, say
10 cm. Take another pair of sticks of
equal lengths, say, 8 cm. Hinge them up
suitably to get a rectangle of length 10 cm
and breadth 8 cm.
This rectangle has been created with
the 4 available measurements.
Now just push along the breadth of
the rectangle. Is the new shape obtained,
still a rectangle (Fig 4.2)? Observe
that the rectangle has now become
a parallelogram. Have you altered the
lengths of the sticks? No! The
measurements of sides remain the same.
Give another push to the newly
obtained shape in a different direction;
what do you get? You again get a
parallelogram, which is altogether different
(Fig 4.3),  yet the four measurements
remain the same.
This shows that 4 measurements of a quadrilateral cannot determine it uniquely .
Can 5 measurements determine a quadrilateral uniquely? Let us go back to the activity!
Practical Geometry
CHAPTER
4
Fig 4.1
Fig 4.2
Fig 4.3
2022-23
58  MATHEMATICS
THINK, DISCUSS AND WRITE
Y ou have constructed a rectangle with
two sticks each of length 10 cm and other
two sticks each of length 8 cm. Now
introduce another stick of length equal to
BD and tie it along BD (Fig 4.4). If you
push the breadth now, does the shape
change? No! It cannot, without making the
figure open. The introduction of the fifth
stick has fixed the rectangle uniquely, i.e.,
there is no other quadrilateral (with the
given lengths of sides) possible now .
Thus, we observe that five measurements can determine a quadrilateral uniquely .
But will any five measurements (of sides and angles) be sufficient to draw a unique
quadrilateral?
Arshad has five measurements of a quadrilateral ABCD. These are AB = 5 cm,
?A = 50°, AC = 4 cm, BD = 5 cm and AD = 6 cm. Can he construct a unique
quadrilateral? Give reasons for your answer.
4.2 Constructing a Quadrilateral
We shall learn how to construct a unique quadrilateral given the following
measurements:
• When four sides and one diagonal are given.
• When two diagonals and three sides are given.
• When two adjacent sides and three angles are given.
• When three sides and two included angles are given.
• When other special properties are known.
Let us take up these constructions one-by-one.
4.2.1 When the lengths of four  sides and a diagonal are given
W e shall explain this construction through an example.
Example 1: Construct a quadrilateral PQRS
where PQ = 4 cm,QR = 6 cm, RS = 5 cm,
PS = 5.5 cm and PR = 7 cm.
Solution: [A rough sketch will help us in
visualising the quadrilateral. W e draw this first and
mark the measurements.] (Fig 4.5)
Fig 4.4
Fig 4.5
2022-23
PRACTICAL GEOMETRY  59
Step 1 From the rough sketch, it is easy to see that ?PQR
can be constructed using SSS construction condition.
Draw ?PQR (Fig 4.6).
Step 2 Now, we have to locate the fourth point S. This ‘S’
would be on the side opposite to Q with reference to
PR. For that, we have two measurements.
S is 5.5 cm away from P . So, with P as centre, draw
an arc of radius 5.5 cm. (The point S is somewhere
on this arc!) (Fig 4.7).
Step 3 S is 5 cm away from R. So with R as centre, draw an arc of radius 5 cm (The
point S is somewhere on this arc also!) (Fig 4.8).
Fig 4.6
Fig 4.7
Fig 4.8
2022-23
60  MATHEMATICS
THINK, DISCUSS AND WRITE
Step 4 S should lie on both the arcs drawn.
So it is the point of intersection of the
two arcs. Mark S and complete PQRS.
PQRS is the required quadrilateral
(Fig 4.9).
(i) We saw that 5 measurements of a quadrilateral can determine a quadrilateral
uniquely. Do you think any five measurements of the quadrilateral can do this?
(ii) Can you draw a parallelogram BATS where BA = 5 cm, AT = 6 cm and
AS = 6.5 cm? Why?
(iii) Can you draw a rhombus ZEAL where ZE = 3.5 cm, diagonal EL = 5 cm? Why?
(iv) A student attempted to draw a quadrilateral PLA Y where PL = 3 cm, LA = 4 cm,
AY = 4.5 cm, PY = 2 cm and LY = 6 cm, but could not draw it. What is
the reason?
[Hint: Discuss it using a rough sketch].
EXERCISE 4.1
1. Construct the following quadrilaterals.
(i) Quadrilateral ABCD. (ii) Quadrilateral JUMP
AB = 4.5 cm JU = 3.5 cm
BC = 5.5 cm UM = 4 cm
CD = 4 cm MP = 5 cm
AD = 6 cm PJ = 4.5 cm
AC = 7 cm PU = 6.5 cm
(iii) Parallelogram MORE (iv) Rhombus BEST
OR = 6 cm BE = 4.5 cm
RE = 4.5 cm ET = 6 cm
EO = 7.5 cm
Fig 4.9
2022-23
PRACTICAL GEOMETRY  61
Fig 4.12
4.2.2  When two diagonals and three sides are given
When four sides and a diagonal were given, we first drew a triangle with the available data
and then tried to locate the fourth point. The same technique is used here.
Example 2: Construct a quadrilateral ABCD, given that BC = 4.5 cm, AD = 5.5 cm,
CD = 5 cm the diagonal AC = 5.5 cm and diagonal BD = 7 cm.
Solution:
Here is the rough sketch of the quadrilateral ABCD
(Fig 4.10). Studying this sketch, we can easily see
that it is possible to draw ? ACD first (How?).
Step 1 Draw ? ACD using SSS
construction (Fig 4.11).
(W e now need to find B at a distance
of 4.5 cm from C and 7 cm from D).
Step 2 With D as centre, draw an arc of radius 7 cm. (B is somewhere
on this arc) (Fig 4.12).
Step 3 With C as centre, draw an arc of
radius 4.5 cm (B is somewhere on
this arc also) (Fig 4.13).
Fig 4.13
Fig 4.11
Fig 4.10
2022-23