NCERT Class 8 Mathematics Ganita Prakash Part 1 Chapter 04 Quadrilaterals PDF Download

Chapter 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.

[Figure: Five quadrilateral and other shapes, See in your textbook]

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.

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

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 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?

Let us first model the structure 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 geometric reasoning (deduction). If that is challenging, try to construct/measure some rectangles.

To find the answers to these 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.

[Figure 1: A rectangle with diagonals AC and BD intersecting at O, See in your textbook]

[Figure 2: A rectangle ABCD with diagonal AC = 8 cm, See in your textbook]

Deduction 1 - What is the length of the other diagonal?

This can be deduced using congruence as follows -

Since ABCD is a rectangle, we have
AB = CD
\( \angle BAD = \angle CDA = 90° \)
AD is common to both triangles.

So, \( \triangle ADC \cong \triangle DAB \) by the SAS congruence condition.

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.

Teacher's Note

When you use congruence to prove properties of rectangles, always start by listing what you know from the definition: all angles are 90°, opposite sides are equal. Then identify which two triangles share a common side or angle. In this case, both triangles share side AD, so that becomes one part of your SAS proof.

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?

[Figure: Rectangle ABCD with diagonals intersecting at O, showing the common side AD, See in your textbook]

The blue angles are equal since they are vertically opposite angles.

[Figure: Rectangle ABCD with diagonals and angles 1 and 2 marked at O, See in your textbook]

In order to show congruence, consider \( \angle 1 \) and \( \angle 2 \). Are they equal?

[Figure: Rectangle ABCD with angles marked, showing Since \( \angle B = 90° \), \( \angle 3 + \angle 1 = 90° \), See in your textbook]

Since \( \angle B = 90° \), \( \angle 3 + \angle 1 = 90° \).

[Figure: Rectangle ABCD with angles marked, showing In \( \triangle BCD \), since \( \angle 3 + \angle 2 + 90 = 180 \), we have \( \angle 3 + \angle 2 = 90° \), See in your textbook]

In \( \triangle BCD \), since \( \angle 3 + \angle 2 + 90 = 180 \), we have \( \angle 3 + \angle 2 = 90° \).

Teacher's Note

To find where the diagonals intersect in a rectangle, you need to compare angles from different triangles. Notice how both \( \angle 3 + \angle 1 \) and \( \angle 3 + \angle 2 \) equal 90°. This means \( \angle 1 = \angle 2 \), which lets you prove two triangles are congruent and show that the diagonals bisect each other.

Key Points

• A rectangle is defined as a quadrilateral with all right angles and opposite sides of equal length.
• The diagonals of a rectangle are always equal in length to each other.
• The diagonals of a rectangle bisect each other, meeting at their midpoints.
• Congruence of triangles is a useful tool to prove properties of rectangles without measuring.

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