Understanding Quadrilaterals Ex 3.1: CBSE Class 8 Maths

What are Polygons? Understanding Curves and Classification

Before we dive deep into quadrilaterals, let's understand the building blocks: curves and polygons. A simple curve is a curve that does not cross itself. Imagine drawing a line without lifting your pencil and without retracing any part of it. If the starting and ending points meet, it's a simple closed curve. All polygons are simple closed curves, but not all simple closed curves are polygons!

A polygon is a simple closed curve made up entirely of line segments. These line segments are called sides, and where two sides meet is called a vertex. For a figure to be a polygon, it must satisfy two conditions:

1. It must be a simple closed curve.
2. It must be made up of only line segments.

Polygons are further classified into convex and concave polygons. A convex polygon is a polygon where all its diagonals lie entirely inside the polygon. Think of a perfect square or a regular hexagon – all connecting lines between non-adjacent vertices stay within the boundary. In contrast, a concave polygon is one where at least a part of one diagonal lies outside the polygon. Imagine a star shape or an arrow pointing inwards – some diagonals would 'poke out' of the shape. Another way to identify a concave polygon is if at least one interior angle is greater than 180 degrees. Understanding these basic distinctions is crucial for identifying and classifying different geometric figures.

Classification of Polygons by Number of Sides

Triangle

A polygon with 3 sides. It is the simplest polygon.

Quadrilateral

A polygon with 4 sides. Examples include squares, rectangles, rhombuses, and trapeziums.

Pentagon

A polygon with 5 sides.

Hexagon

A polygon with 6 sides.

Heptagon (or Septagon)

A polygon with 7 sides.

Octagon

A polygon with 8 sides.

Nonagon (or Enneagon)

A polygon with 9 sides.

Decagon

A polygon with 10 sides.

Identifying Different Types of Curves and Polygons

• Let's look at various figures and categorize them. Imagine the following diagrams (you can sketch them as we go!): Figure 1: A freely drawn wavy line that doesn't cross itself. Type: Simple Curve. Reason: It does not cross itself, but its endpoints do not meet. Figure 2: A square. Type: Simple Curve, Simple Closed Curve, Polygon, Convex Polygon. Reason: It doesn't cross itself, its endpoints meet, it's made of line segments, and all its diagonals lie inside. Figure 3: A figure like a crescent moon or a capital 'C'. Type: Simple Curve, Simple Closed Curve. Reason: It doesn't cross itself, its endpoints meet, but it's not made entirely of line segments (it has a curved boundary). Figure 4: Two overlapping circles. Type: Not a simple curve. Reason: The curves cross each other. Figure 5: A five-pointed star. Type: Simple Closed Curve, Polygon, Concave Polygon. Reason: It doesn't cross itself, it's closed, made of line segments, but if you draw a diagonal from one 'inner' vertex to another, part of it will be outside the star, or it has interior angles greater than 180 degrees.

How to Count Diagonals in a Polygon

1. Understand Diagonals — A diagonal is a line segment connecting two non-consecutive vertices of a polygon. 'Non-consecutive' means they are not next to each other along a side.
2. Identify Vertices — First, identify all the vertices (corners) of your polygon. Let's say a polygon has 'n' vertices.
3. Draw Diagonals from One Vertex (Visual Method) — Pick any one vertex. From this vertex, you can draw a diagonal to every other vertex EXCEPT itself and its two adjacent (next-door) vertices. So, from one vertex, you can draw (n - 3) diagonals. For example, in a quadrilateral (n=4), from one vertex, you can draw (4-3) = 1 diagonal.
4. Consider All Vertices and Avoid Duplicates (Formula Derivation) — If you do this for all 'n' vertices, you might think there are n * (n-3) diagonals. However, this counts each diagonal twice (once from each endpoint). For example, the diagonal from A to C is the same as C to A. Therefore, we divide by 2. The formula to find the number of diagonals in a polygon with 'n' sides (or 'n' vertices) is: Number of diagonals = n(n-3)/2.
5. Example: Counting Diagonals in a Hexagon (6 sides) — Using the formula: n = 6. Number of diagonals = 6 (6 - 3) / 2 = 6 3 / 2 = 18 / 2 = 9 diagonals. You can try drawing a hexagon and verifying this by connecting all non-adjacent vertices.

Exam Tip: Distinguishing Polygon Types and Diagonals

When identifying polygons, always check both conditions: is it a simple closed curve, AND is it made only of line segments? Many students confuse figures with curved parts (like a circle or a semi-circle) with polygons.

For convex vs. concave, remember: if any part of any diagonal lies outside the polygon, it's concave. A quick check is to look for 'dents' or 'inward turns' in the polygon; these usually indicate a concave shape. Also, if there's an interior angle greater than 180 degrees, it's concave.

When counting diagonals, especially for smaller polygons, draw them carefully. For larger polygons, use the formula n(n-3)/2 to save time and ensure accuracy, where 'n' is the number of sides/vertices. Don't forget to divide by 2!

Practice Questions with Solutions

• Q: Classify the following figures as (a) Simple curve (b) Simple closed curve (c) Polygon (d) Convex polygon (e) Concave polygon. (Imagine a picture of a regular pentagon) A: Step 1: Analyze the figure. It is drawn without lifting the pen and without crossing itself. Step 2: Its endpoints meet, forming a closed shape. Step 3: It is made entirely of five straight line segments. Step 4: All its diagonals lie completely inside the figure. Final answer: (a) Simple curve, (b) Simple closed curve, (c) Polygon, (d) Convex polygon.
• Q: Is a figure formed by joining three non-collinear points a polygon? If yes, what kind? A: Step 1: Non-collinear points mean they do not lie on the same straight line. Joining three such points forms a triangle. Step 2: A triangle is a simple closed curve made of three line segments. Step 3: All its diagonals (it has none, as n=3, so 3(3-3)/2 = 0) would trivially be inside, and all angles are less than 180 degrees. Final answer: Yes, it is a polygon. Specifically, it is a triangle, which is also a convex polygon.
• Q: How many diagonals does a hexagon have? Show your calculation. A: Step 1: A hexagon has 'n' = 6 sides. Step 2: The formula for the number of diagonals in a polygon is n(n-3)/2. Step 3: Substitute n=6 into the formula: 6 (6-3) / 2. Step 4: Calculate: 6 3 / 2 = 18 / 2 = 9. Final answer: A hexagon has 9 diagonals.
• Q: A shape has one interior angle greater than 180 degrees. What type of polygon is it? A: Step 1: Recall the definitions of convex and concave polygons. Step 2: A convex polygon has all interior angles less than 180 degrees. Step 3: A concave polygon has at least one interior angle greater than 180 degrees (or at least one diagonal partially outside). Final answer: It is a concave polygon.
• Q: Draw a figure that is a simple closed curve but not a polygon. A: Step 1: A simple closed curve means it doesn't cross itself and its starting and ending points meet. Step 2: Not a polygon means it cannot be made entirely of line segments; it must have at least one curved part. Step 3: Sketch a circle, an oval, or a crescent shape (like a 'C' with its ends joined by a curve). Final answer: (Student should draw) A circle or an oval are perfect examples. (E.g., imagine a circular shape).

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