NCERT Class 9 Ganita Manjari Part 2 Chapter 12 Quadrilaterals PDF Download

Chapter 12: Quadrilaterals

Think and Reflect

Can we use any given quadrilateral to tile the plane? If not, which quadrilaterals can be used and which cannot?

Recall that tiling the plane means covering it with copies of the given shape or shapes without gaps or overlaps. Tiling the plane with a rectangle is easy, but it seems difficult with an irregular quadrilateral. We will answer the above question at the end of the chapter (can you do it now?) and understand the answer using what we learn in this chapter. However, to start with, we will discuss a simple new example of a tiling.

Imagine that tiling with rectangles is done using a grid of two sets of long wooden sticks, one set horizontal and the other set vertical. See Fig. 12.1A.

[Figure 12.1A: A child holding a grid of intersecting sticks arranged in rectangles, and Fig. 12.1B: The same grid pushed to show parallelograms, See in your textbook]

These sticks are pinned together at each intersection point such that they are free to rotate about those points. Now suppose you pick up this grid. You hold the bottom stick fixed with one hand and push the vertical side with the other. See Fig. 12.1B. Each set of parallel lines will stay parallel but the angle between the two sets will no longer be \( 90° \). Thus we see that the plane can be tiled using copies of any fixed parallelogram.

In this chapter, we will ask and answer several questions about quadrilaterals, paying particular attention to parallelograms. We will investigate their properties, examine ways to recognise them, and see important applications in geometry. At the end, we will answer the question above about whether one can tile the plane with a given quadrilateral.

12.1 What Exactly is a Quadrilateral?

Think and Reflect

Informally, a quadrilateral is a figure with four straight sides, as in the first figure ABCD in Fig. 12.2 below. But consider the other six figures in Fig. 12.2: the five plane figures NOPE, SILY, DART, CUTS, OPENS and the non-planar BENT. Should we call all these figures quadrilaterals? If you answer 'no' for any of them, how will you define a quadrilateral so that such a figure is excluded? As you can see, some care is needed to precisely define what we think of as a quadrilateral.

[Figure 12.2: Seven figures with four vertices labeled: ABCD (a simple quadrilateral), NOPE, SILY, DART, CUTS, OPENS, and BENT showing various configurations, See in your textbook]

To prepare, let us first consider how we can define a triangle. Let A, B and C be three points. Can we say that \( \triangle ABC \) consists of points on the three line segments AB, BC, CA? Yes, provided A, B, C are not collinear. Otherwise, we just get a line segment: see Fig. 12.3.

[Figure 12.3: Not a triangle - three collinear points A, B, C on a line, See in your textbook]

When A, B, C are non-collinear, we call them the vertices of \( \triangle ABC \) and the three segments AB, BC and CA the sides of \( \triangle ABC \).

Think and Reflect

Can we similarly define a quadrilateral ABCD?

Take any 4 distinct non-collinear points A, B, C and D. Will segments AB, BC, CD and DA always form a quadrilateral as we visualise it? See figures NOPE, SILY, DART, CUTS, OPENS and BENT in Fig. 12.2 again. Do you see, OPENS, that there are some subtle new issues to consider?

First, to rule out examples like NOPE and SILY, no three vertices should be collinear. Otherwise, the two adjacent sides formed by the three collinear vertices will be along the same line, giving a triangle like NOPE or a figure with overlapping sides like in SILY.

Second, to rule out examples like BENT, all four vertices should be in the same plane. Four-sided figures like BENT are called non-planar quadrilaterals. We will study only planar quadrilaterals (except in a few exercises).

Third, do we want to call CUTS a quadrilateral? The answer depends on what we decide! Ordinarily, CUTS is not considered a quadrilateral. If we want to include it, we call it a self-intersecting quadrilateral. Can you come up with a definition that rules out self-intersecting quadrilaterals like CUTS? Answer: we should require that all the points of the quadrilateral, other than the vertices, must lie on exactly one side. Can you see how to rule out figures like OPENS?

Putting together all three requirements, we get the following definition.

Definition 1

Suppose A, B, C, D are four distinct points in a plane. Then the points on the 4 segments AB, BC, CD and DA are said to form quadrilateral ABCD if every such point other than A, B, C, and D lies on exactly one of these four segments.

Teacher's Note

The key phrase is "exactly one of these four segments." This rules out self-intersecting quadrilaterals where a point could lie on two different sides. When you draw a quadrilateral, make sure the sides do not cross each other except at the vertices - that's what "exactly one" ensures.

Key Points

• A quadrilateral is a figure with four vertices and four sides, where all vertices lie in the same plane and no three vertices are collinear.
• In a proper (non-self-intersecting) quadrilateral, every point on the sides other than the vertices lies on exactly one of the four segments.
• Parallelograms can tile the plane by fitting together without gaps or overlaps, as shown by pushing and rotating a grid of intersecting lines.

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