This MCQ module is based on: Rectangles, Squares and Diagonals
TOPIC 11 OF 22
Rectangles, Squares and Diagonals
🎓 Class 8
Mathematics
CBSE
Theory
Ch 4 — Quadrilaterals
⏱ ~35 min
🌐 Language: [gtranslate]
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This mathematics assessment will be based on: Rectangles, Squares and Diagonals
Targeting Class 8 level in Geometry, with Basic difficulty.
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4. Quadrilaterals — Introduction
In this chapter, we will study some interesting types of four-sided figures and solve some 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.
Figures (i), (ii), (iii) are quadrilaterals; (iv) has five sides, (v) has three — not quadrilaterals.
The angles of a quadrilateral are the angles between its sides, as marked in figures (i), (ii), and (iii). We will start with the most familiar quadrilaterals — rectangles and squares.
4.1 Rectangles and Squares
We know that rectangles exist. Let us define them formally.
Definition — 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 strip of 8 cm long. What should be the length of the other strip? Where should they both be joined? Let us first model the structure (Fig. 1). The strips can be modelled as line segments. They act as the diagonals? of the quadrilateral formed by their endpoints. For the quadrilateral to be a rectangle, we need to answer the following questions:
- What is the length of the other diagonal?
- What is the point of intersection of the two diagonals?
- What should be the angle between the two diagonals?
Rectangle ABCD with diagonals AC and BD intersecting at O.
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, ∠BAD = ∠CDA = 90°. AD is common to both triangles. So, \(\triangle ADB \cong \triangle DAC\) by the SAS congruence condition.
Therefore, AC = BD, since they are corresponding parts of congruent triangles. This shows that the diagonals of a rectangle are equal. So the other diagonal must also be 8 cm, the same length. You can verify this property by constructing/measuring more 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?
Vertically opposite angles ∠1 = ∠2 at O.
The blue angles are equal since they are vertically opposite angles. In order to show congruence, consider ∠1 and ∠2. Are they equal? Since ∠B = 90°, ∠3 + ∠1 = 90°. In ΔBCA, since ∠3 + ∠2 + ∠C = 180°, we have ∠3 + ∠2 = 90°. So, ∠1 = ∠2 (= 90° − ∠3).
Thus, by the AAS condition, \(\triangle AOB \cong \triangle COD\). Hence OC = OA and OD = OB, so O is the midpoint of AC and of BD. This shows that the diagonals of a rectangle always intersect at their midpoints.
When the diagonals cross at their midpoints, we say that each bisects the 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 ΔAOD ≅ ΔCOB? AO = OC (proved above); ∠AOD = ∠COB (vertically opposite angles); AD = CB (opposite sides of a rectangle). Yes — by SAS.
Deduction 3 — What are the angles between the diagonals?
Let us check what quadrilateral we get if we draw the two diagonals such that their lengths are equal, they bisect each other at an arbitrary angle, say 60°, between them.
Angles at O: 60° and 120° (pair of vertically opposite angles).
In ΔAOB, we have \(a + a + 60° = 180°\) (angle sum of a triangle). Therefore \(2a = 120°\), so \(a = 60°\). Similarly, we can find the values of all the other angles.
Can we now identify what type of quadrilateral ABCD is? Notice that its angles all add up to 90° (30° + 60°). What can we say about its sides?
We can see that \(\triangle AOB \cong \triangle COD\) and \(\triangle AOD \cong \triangle COB\). Hence AB = CD, and AD = CB, since they are corresponding parts of congruent triangles. Therefore, ABCD is a quadrilateral satisfying the definition of a rectangle.
Key Property
In a rectangle, the diagonals are equal in length and bisect each other. Conversely, if the diagonals of a quadrilateral are equal in length and bisect each other, then the quadrilateral is a rectangle.
Activity: Construct a Rectangle from its Diagonals
L3 Apply
Materials: Two sticks (one 8 cm, one 8 cm), string, pin/board, ruler, protractor.
Predict: If the two sticks are pinned at their midpoints and you rotate them, what shapes can you make?
- Pin the two 8 cm sticks at their midpoints, so each bisects the other.
- Connect the four endpoints with string — the figure is a quadrilateral.
- Rotate one stick. Measure the sides: are opposite sides equal?
- At what angle between the sticks do you get: (a) a rectangle, (b) a square?
Any angle of intersection produces a rectangle (since diagonals are equal & bisect each other). When the sticks are perpendicular (90°), the rectangle becomes a square.
Figure it Out
Q1. Can you find the value of α (the angle opposite to the 120° angle between diagonals in the rhombus ABCD)?
The two angles on a straight line sum to 180°: α + 120° = 180°, so α = 60°.
Q2. Will ABCD remain a rectangle if the angles between the diagonals are changed? Can we generalise this?
Yes, ABCD remains a rectangle as long as the diagonals are equal and bisect each other — regardless of the angle between them. If the angle becomes 90°, ABCD becomes a square.
Competency-Based Questions
Scenario: An architect is designing a rectangular window. She tells the carpenter: "Make two diagonal support rods of equal length, joined at their midpoints." The carpenter uses two rods of 10 cm each.
Q1. The rods are fixed at 90° to each other at the midpoint. What quadrilateral do the endpoints form?
L3 ApplyA square — diagonals equal, bisect each other, perpendicular.
Q2. Analyse: if the rods are fixed at 70° instead, is the shape still a rectangle? Justify.
L4 AnalyseYes. Equal diagonals bisecting each other always give a rectangle, regardless of the angle between the diagonals. The sides change in length, but angles remain 90°.
Q3. Evaluate: the carpenter accidentally uses rods of 10 cm and 8 cm. Can the endpoints still form a rectangle?
L5 EvaluateNo. Rectangles require equal diagonals. Unequal diagonals give, at most, a parallelogram (if they bisect each other) but not a rectangle.
Q4. Create a procedure another carpenter could use, with only a measuring tape, to verify that a completed frame is exactly a rectangle.
L6 CreateProcedure: (1) measure opposite sides — must be equal; (2) measure the two diagonals — must be equal. If both conditions hold, the frame is a rectangle. (No protractor needed.)
Assertion–Reason Questions
A: The diagonals of a rectangle are equal in length.
R: \(\triangle ADB \cong \triangle DAC\) by SAS.
R: \(\triangle ADB \cong \triangle DAC\) by SAS.
(a) — The congruence directly gives AC = BD.
A: The diagonals of a rectangle always intersect at right angles.
R: Rectangles have four right angles at their vertices.
R: Rectangles have four right angles at their vertices.
(d) — A is false: diagonals of a rectangle are perpendicular only in the special case of a square. R is true but does not imply A.
Frequently Asked Questions
What are the properties of a rectangle?
A rectangle has four right angles, opposite sides equal and parallel, and two equal diagonals that bisect each other. Its diagonals divide it into two congruent right triangles. NCERT Class 8 Ganita Prakash Part 1 Chapter 4 lists these properties.
How is a square different from a rectangle?
A square is a special rectangle where all four sides are equal. Every square is a rectangle, but not every rectangle is a square. Both have four right angles; the square adds the equal-sides condition. NCERT Class 8 Chapter 4 clarifies this.
What is special about a square's diagonals?
A square's diagonals are equal in length, bisect each other at right angles, and bisect the corner angles (each 45°). These properties follow from the four equal sides and four right angles. NCERT Class 8 Ganita Prakash Part 1 Chapter 4.
How do you find the diagonal of a rectangle?
Use Pythagoras' theorem: if the rectangle has sides a and b, the diagonal d satisfies d^2 = a^2 + b^2, so d = sqrt(a^2 + b^2). For a 3 x 4 rectangle, d = sqrt(9 + 16) = 5. NCERT Class 8 Chapter 4 applies this.
What is a quadrilateral?
A quadrilateral is a closed four-sided polygon with four vertices and four angles. Rectangles, squares, parallelograms, rhombuses, and trapeziums are all quadrilaterals. NCERT Class 8 Ganita Prakash Part 1 Chapter 4 opens with this definition.
Why do we study diagonals?
Diagonals reveal symmetry and relationships within quadrilaterals - equal length, perpendicularity, bisection - which define each shape. Measuring diagonals also gives a way to calculate areas and perimeters indirectly. NCERT Class 8 Chapter 4 stresses this.
Frequently Asked Questions — Quadrilaterals
What is Rectangles, Squares and Diagonals in NCERT Class 8 Mathematics?
Rectangles, Squares and Diagonals is a key concept covered in NCERT Class 8 Mathematics, Chapter 4: Quadrilaterals. This lesson builds the student's foundation in the chapter by explaining the core ideas with worked examples, definitions, and step-by-step methods aligned to the CBSE curriculum.
How do I solve problems on Rectangles, Squares and Diagonals step by step?
To solve problems on Rectangles, Squares and Diagonals, follow the NCERT method: identify the given quantities, choose the relevant formula or theorem, substitute values carefully, and simplify. Class 8 exercises gradually increase in difficulty — start with solved NCERT examples before attempting exercise questions, and always verify your answer by substitution or diagram.
What are the most important formulas for Chapter 4: Quadrilaterals?
The essential formulas of Chapter 4 (Quadrilaterals) are listed in the chapter summary and highlighted throughout the lesson in formula boxes. Memorise them and practise at least 2–3 problems per formula. CBSE board exams frequently test direct application as well as combined use of multiple formulas from this chapter.
Is Rectangles, Squares and Diagonals important for the Class 8 board exam?
Rectangles, Squares and Diagonals is part of the NCERT Class 8 Mathematics syllabus and appears in CBSE board exams. Questions typically include short-answer, long-answer, and competency-based items. Review the NCERT examples, exercise questions, and previous-year board problems on this topic to prepare confidently.
What mistakes should students avoid in Rectangles, Squares and Diagonals?
Common mistakes in Rectangles, Squares and Diagonals include skipping steps, misapplying formulas, sign errors, and losing track of units. Write each step clearly, double-check algebraic manipulations, and re-read the question after solving to verify that your answer matches what was asked.
Where can I find more NCERT practice questions on Rectangles, Squares and Diagonals?
End-of-chapter NCERT exercises for Rectangles, Squares and Diagonals cover all difficulty levels tested in CBSE exams. After completing them, try the examples again without looking at the solutions, attempt the NCERT Exemplar questions for Chapter 4, and solve at least one previous-year board paper to consolidate your understanding.
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