4 QUADRILATERALS
In this chapter, we will study some interesting types of four-sided
figures and solve problems based on them. Such figures are commonly
known as quadrilaterals. The word ‘quadrilateral’ is derived from Latin
words — quadri meaning four, and latus referring to sides.
Observe the following figures.
(i) (ii) (iii)
(iv) (v)
Figs. (i), (ii), and (iii) are quadrilaterals, and the others are not. Why?
The angles of a quadrilateral are the angles between its sides, as
marked in Figs. (i), (ii), and (iii).
We will start with the most familiar quadrilaterals — rectangles and
squares.
Reprint 2026-27
, Quadrilaterals
4.1 Rectangles and Squares
We know what rectangles are. Let us define them.
Rectangle: A rectangle is a quadrilateral in which —
(i) The angles are all right angles (90°), and
(ii) The opposite sides are of equal length.
The definition precisely states the conditions a quadrilateral has to
satisfy to be called a rectangle.
Are there other ways to define a rectangle?
Let us consider the following problem related to the construction of
rectangles.
A Carpenter’s Problem
B C
A carpenter needs to put together
two thin strips of wood, as shown
in Fig. 1, so that when a thread is
passed through their endpoints, it
O
forms a rectangle.
She already has one 8 cm long
strip. What should be the length of
the other strip? Where should they
both be joined?
A D
Let us first model the structure Fig. 1
that the carpenter has to make. The
strips can be modelled as line segments. They are the diagonals of the
quadrilateral formed by their endpoints. For the quadrilateral to be a
rectangle, we need to answer the following questions —
1. What is the length of the other diagonal?
2. What is the point of intersection of the two diagonals?
3. What should the angle be between the diagonals?
Let us answer these questions using B C
geometric reasoning (deduction). If
that is challenging, try to construct/
measure some rectangles.
To find the answers to these AC = 8 cm
questions, let us suppose that we
have placed the diagonals such that
their endpoints form the vertices of
a rectangle, as shown in Fig. 2. A D
Fig. 2
83
Reprint 2026-27
, Ganita Prakash | Grade 8
Deduction 1 — What is the length of the other diagonal?
This can be deduced using congruence as follows —
B C
Since ABCD is a rectangle, we have
AB = CD
∠BAD = ∠CDA = 90°
AD is common to both triangles.
So, ∆ADC ≅ ∆DAB by the SAS congruence
condition. A D
Common side
Therefore, AC = BD, since they are corresponding parts of congruent
triangles. This shows that the diagonals of a rectangle always have the
same length.
So the other diagonal must also be 8 cm long. You can verify this
property by constructing/measuring some rectangles.
Deduction 2 — What is the point of intersection of the two diagonals?
This can also be found using congruence. Since we need to know the
relation between OA and OC, and OB and OD, which two triangles of the
rectangle ABCD should we consider?
B C B C
1
O O
2
A D A D
The blue angles are equal since In order to show congruence,
they are vertically opposite consider ∠1 and ∠2. Are they
angles. equal?
B C B C
3 3
1
O O
2
A D A D
Since ∠B = 90°, In ∆BCD, since
84 ∠3 + ∠1 = 90°. ∠3 + ∠2 + 90 = 180,
we have ∠3 + ∠2 = 90°.
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, Quadrilaterals
So, ∠1 = ∠2 (= 90° – ∠3).
Thus, by the AAS condition for congruence, ∆AOB ≅ ∆COD.
Hence OA = OC and OB = OD, since they are corresponding parts of
congruent triangles. So, O is the midpoint of AC and BD.
This shows that the diagonals of a rectangle always intersect at
their midpoints.
Therefore, to get a rectangle, the diagonals must be drawn so that
they are equal and intersect at their midpoints.
When the diagonals cross at their midpoints, we say that the diagonals bisect
each other. Bisecting a quantity means dividing it into two equal parts.
Verify this property by constructing some rectangles and measuring
their diagonals and the points of intersection.
Can the following equalities be used to establish that ∆AOD ≅ ∆COB? Math
AO = CO (proved above) Talk
∠AOB = ∠COD (vertically opposite angles)
AD = CB
Deduction 3 — What are the angles between the diagonals?
Let us check what quadrilateral we get if we draw the A B
two diagonals such that their lengths are equal, they
bisect each other and have an arbitrary angle, say
60°, between them as shown in the figure to the right.
60°
Can you find all the remaining angles?
O
A B
D C
60°
O We can find the remaining angles between the
120° 120° diagonals using our understanding of vertically
opposite angles and linear pairs.
60°
A B
a a
D C
60°
In ∆AOB, since OA = OB, the angles opposite them 120° 120°
O
are equal, say a. 60°
Can you find the value of a?
D C 85
Reprint 2026-27
In this chapter, we will study some interesting types of four-sided
figures and solve problems based on them. Such figures are commonly
known as quadrilaterals. The word ‘quadrilateral’ is derived from Latin
words — quadri meaning four, and latus referring to sides.
Observe the following figures.
(i) (ii) (iii)
(iv) (v)
Figs. (i), (ii), and (iii) are quadrilaterals, and the others are not. Why?
The angles of a quadrilateral are the angles between its sides, as
marked in Figs. (i), (ii), and (iii).
We will start with the most familiar quadrilaterals — rectangles and
squares.
Reprint 2026-27
, Quadrilaterals
4.1 Rectangles and Squares
We know what rectangles are. Let us define them.
Rectangle: A rectangle is a quadrilateral in which —
(i) The angles are all right angles (90°), and
(ii) The opposite sides are of equal length.
The definition precisely states the conditions a quadrilateral has to
satisfy to be called a rectangle.
Are there other ways to define a rectangle?
Let us consider the following problem related to the construction of
rectangles.
A Carpenter’s Problem
B C
A carpenter needs to put together
two thin strips of wood, as shown
in Fig. 1, so that when a thread is
passed through their endpoints, it
O
forms a rectangle.
She already has one 8 cm long
strip. What should be the length of
the other strip? Where should they
both be joined?
A D
Let us first model the structure Fig. 1
that the carpenter has to make. The
strips can be modelled as line segments. They are the diagonals of the
quadrilateral formed by their endpoints. For the quadrilateral to be a
rectangle, we need to answer the following questions —
1. What is the length of the other diagonal?
2. What is the point of intersection of the two diagonals?
3. What should the angle be between the diagonals?
Let us answer these questions using B C
geometric reasoning (deduction). If
that is challenging, try to construct/
measure some rectangles.
To find the answers to these AC = 8 cm
questions, let us suppose that we
have placed the diagonals such that
their endpoints form the vertices of
a rectangle, as shown in Fig. 2. A D
Fig. 2
83
Reprint 2026-27
, Ganita Prakash | Grade 8
Deduction 1 — What is the length of the other diagonal?
This can be deduced using congruence as follows —
B C
Since ABCD is a rectangle, we have
AB = CD
∠BAD = ∠CDA = 90°
AD is common to both triangles.
So, ∆ADC ≅ ∆DAB by the SAS congruence
condition. A D
Common side
Therefore, AC = BD, since they are corresponding parts of congruent
triangles. This shows that the diagonals of a rectangle always have the
same length.
So the other diagonal must also be 8 cm long. You can verify this
property by constructing/measuring some rectangles.
Deduction 2 — What is the point of intersection of the two diagonals?
This can also be found using congruence. Since we need to know the
relation between OA and OC, and OB and OD, which two triangles of the
rectangle ABCD should we consider?
B C B C
1
O O
2
A D A D
The blue angles are equal since In order to show congruence,
they are vertically opposite consider ∠1 and ∠2. Are they
angles. equal?
B C B C
3 3
1
O O
2
A D A D
Since ∠B = 90°, In ∆BCD, since
84 ∠3 + ∠1 = 90°. ∠3 + ∠2 + 90 = 180,
we have ∠3 + ∠2 = 90°.
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, Quadrilaterals
So, ∠1 = ∠2 (= 90° – ∠3).
Thus, by the AAS condition for congruence, ∆AOB ≅ ∆COD.
Hence OA = OC and OB = OD, since they are corresponding parts of
congruent triangles. So, O is the midpoint of AC and BD.
This shows that the diagonals of a rectangle always intersect at
their midpoints.
Therefore, to get a rectangle, the diagonals must be drawn so that
they are equal and intersect at their midpoints.
When the diagonals cross at their midpoints, we say that the diagonals bisect
each other. Bisecting a quantity means dividing it into two equal parts.
Verify this property by constructing some rectangles and measuring
their diagonals and the points of intersection.
Can the following equalities be used to establish that ∆AOD ≅ ∆COB? Math
AO = CO (proved above) Talk
∠AOB = ∠COD (vertically opposite angles)
AD = CB
Deduction 3 — What are the angles between the diagonals?
Let us check what quadrilateral we get if we draw the A B
two diagonals such that their lengths are equal, they
bisect each other and have an arbitrary angle, say
60°, between them as shown in the figure to the right.
60°
Can you find all the remaining angles?
O
A B
D C
60°
O We can find the remaining angles between the
120° 120° diagonals using our understanding of vertically
opposite angles and linear pairs.
60°
A B
a a
D C
60°
In ∆AOB, since OA = OB, the angles opposite them 120° 120°
O
are equal, say a. 60°
Can you find the value of a?
D C 85
Reprint 2026-27
