NCERT Solutions Class 8 Maths Chapter 3 Exercise 3.4: Understanding Quadrilaterals

The Family Tree of Quadrilaterals

Every special quadrilateral inherits properties from its mathematical ancestors. A general quadrilateral is simply a closed four-sided polygon. If it acquires at least one pair of parallel sides, it becomes a trapezium. If it gets two pairs of parallel sides, it steps up to become a parallelogram. From the parallelogram, we branch out into two special shapes: the rectangle (which has four right angles) and the rhombus (which has four equal sides). Now, what happens when a shape possesses both the equal-angle property of a rectangle AND the equal-side property of a rhombus? It becomes a square! Therefore, a square is a highly specialized quadrilateral that is simultaneously a parallelogram, a rectangle, and a rhombus. Understanding this logical nesting is the secret to solving Exercise 3.4 correctly without memorization.

Comparing Rhombus vs Rectangle Properties

Worked Examples: Reasoning and Concept Analysis

• Example 1: State whether the statement is True or False: 'All rectangles are squares.' Reasoning: A square must have all four sides of equal length. While a rectangle has four right angles and equal diagonals, it does not necessarily have all four sides equal. Since some rectangles have unequal adjacent sides, the statement is False.
• Example 2: State whether the statement is True or False: 'All rhombuses are parallelograms.' Reasoning: By definition, a parallelogram is a quadrilateral with opposite sides parallel. A rhombus has both pairs of opposite sides parallel (since it is a special type of parallelogram with equal sides). Therefore, every single rhombus satisfies the criteria to be a parallelogram. The statement is True.
• Example 3: Identify all the quadrilaterals that have four sides of equal length. Reasoning: Step 1: Recall the definitions of special quadrilaterals. Step 2: A square is defined as having four equal sides and four right angles. Step 3: A rhombus is defined as a parallelogram with four equal sides. Step 4: Other quadrilaterals like rectangles, parallelograms, and trapeziums do not require all four sides to be equal. Final Answer: The quadrilaterals with four sides of equal length are Rhombus and Square.

Exam Tip: The Diagonal Checklist

In CBSE board exams, questions on diagonal properties of quadrilaterals are extremely common. To avoid getting confused, memorize this simple three-question checklist:

1. Do the diagonals bisect each other? Yes, for all parallelograms (including rectangle, rhombus, and square).
2. Are they equal in length? Only for Rectangles and Squares.
3. Are they perpendicular to each other? Only for Rhombuses, Squares, and Kites.

Common Mistake: Students often think that because a rectangle has equal diagonals, a rhombus must have them too. Remember, a rhombus is a 'squashed' square; its diagonals are always perpendicular, but they are unequal in length!

Practice Questions with Solutions

• Q: Explain why a square is a convex quadrilateral. A: Step 1: Recall the definition of a convex quadrilateral. A quadrilateral is convex if all its interior angles are less than 180° and both of its diagonals lie entirely in its interior. Step 2: For a square, each of the four interior angles is exactly 90° (which is less than 180°). Step 3: If you draw both diagonals of a square, they intersect inside the square, meaning the diagonals lie completely in its interior. Final answer: Since all interior angles are less than 180° and both diagonals lie inside the figure, a square is a convex quadrilateral.
• Q: Name the quadrilaterals whose diagonals are perpendicular bisectors of each other. A: Step 1: Identify what 'perpendicular bisectors' means. It means the diagonals cut each other into two equal parts (bisect) and meet at a 90-degree angle (perpendicular). Step 2: Analyze a rhombus. Its diagonals bisect each other at right angles. So, Rhombus is one. Step 3: Analyze a square. Since a square is a type of rhombus, its diagonals also bisect each other at 90 degrees. So, Square is another. Step 4: Analyze a kite. Its diagonals are perpendicular, but only one diagonal bisects the other, so it is not a mutual perpendicular bisector. Final answer: Rhombus and Square are the quadrilaterals whose diagonals are perpendicular bisectors of each other.
• Q: State whether the statement is True or False and explain: 'All squares are trapeziums.' A: Step 1: Define a trapezium. A trapezium is a quadrilateral with at least one pair of parallel sides. Step 2: Examine a square. A square has two pairs of parallel opposite sides. Step 3: Since having two pairs of parallel sides automatically satisfies the requirement of having 'at least one pair' of parallel sides, every square qualifies as a trapezium. Final answer: True. Since a square has opposite sides parallel, it is also a trapezium.
• Q: Name the quadrilateral that is both equilateral (all sides equal) and equiangular (all angles equal). A: Step 1: Identify the shape with all equal sides (equilateral quadrilateral). This can be a rhombus or a square. Step 2: Identify the shape with all equal angles (equiangular quadrilateral). This can be a rectangle or a square. Step 3: Find the intersection of both properties. The shape that has both all equal sides and all equal angles is a square (each angle is 90° and all four sides are equal). Final answer: The square is the quadrilateral that is both equilateral and equiangular.

Frequently Asked Questions