Sample Paper for Class 8 Mathematics (Ganit Prakash – Chapter 4: Quadrilaterals

Sample Paper for Class 8 Mathematics (Ganit Prakash – Chapter 4: Quadrilaterals

🧮 Sample Paper 

Chapter 4: Quadrilaterals 

(चतुर्भुज)

Subject: Mathematics (गणित)
Class: 8
Time: 1 hour 30 minutes (1 घंटा 30 मिनट)

Full Marks: 40

भाग – A : 

वस्तुनिष्ठ प्रश्न (Objective Type Questions)

(प्रत्येक प्रश्न 1 अंक का है)

01. किसी चतुर्भुज के चारों कोणों का योगफल होता है –

The sum of all angles of a quadrilateral is –

(A) 180°  (B) 270°  (C) 360°  (D) 90°

02. वह चतुर्भुज जिसके विपरीत भुजाएँ समान और समान्तर होती हैं, कहलाता है –

(A) वर्ग (B) आयत (C) समान्तर चतुर्भुज (D) समचतुर्भुज

A quadrilateral whose opposite sides are equal and parallel is called –

(A) Square (B) Rectangle (C) Parallelogram (D) Rhombus

03. आयत की विकर्ण रेखाएँ –

(A) समान होती हैं (B) असमान होती हैं (C) लम्ब होती हैं (D) समान और लम्ब दोनों होती हैं

The diagonals of a rectangle are –

(A) Equal (B) Unequal (C) Perpendicular (D) Equal and perpendicular both

04. किसी समचतुर्भुज की सभी भुजाएँ समान होती हैं, पर उसके कोण –

(A) सभी समकोण (B) दो समकोण (C) समान नहीं होते (D) तीक्ष्ण होते हैं

All sides of a rhombus are equal, but its angles are –

(A) All right angles (B) Two right angles (C) Not equal (D) Acute

05. वर्ग एक –

(A) समान्तर चतुर्भुज है। (B) आयत है । (C) समचतुर्भुज है। (D) उपरोक्त सभी

A square is a –

(A) Parallelogram (B) Rectangle (C) Rhombus (D) All of these

06. यदि किसी समान्तर चतुर्भुज की एक भुजा 8 सेमी और दूसरी 6 सेमी है, तो उसका परिमाप क्या होगा?

(A) 14 सेमी (B) 24 सेमी (C) 28 सेमी (D) 36 सेमी

If one side of a parallelogram is 8 cm and the adjacent side is 6 cm, then its perimeter is –

(A) 14 cm (B) 24 cm (C) 28 cm (D) 36 cm

07. कौन सा चतुर्भुज न तो समान्तर चतुर्भुज है, न आयत, न समचतुर्भुज, लेकिन उसके केवल एक जोड़ी भुजाएँ समान्तर होती हैं?

(A) पतंग (B) वर्ग (C) समलंब (Trapezium) (D) समान्तर चतुर्भुज

Which quadrilateral has only one pair of parallel sides?

(A) Kite (B) Square (C) Trapezium (D) Parallelogram

08. समलंब के विकर्ण सामान्यतः –

(A) समान होते हैं (B) लम्बवत होते हैं (C) असमान होते हैं (D) समान और लम्ब दोनों

The diagonals of a trapezium are generally –

(A) Equal (B) Perpendicular (C) Unequal (D) Equal and perpendicular both

भाग – B : 

लघु उत्तर प्रश्न (Short Answer Type)

(प्रत्येक प्रश्न 2 अंक का है)

09. एक चतुर्भुज के तीन कोण क्रमशः 70°, 80°, और 90° हैं। चौथा कोण ज्ञात करो।

The three angles of a quadrilateral are 70°, 80°, and 90°. Find the fourth angle.

10. समान्तर चतुर्भुज की विशेषताएँ लिखो।

Write any two properties of a parallelogram.

11. आयत और समचतुर्भुज में क्या अंतर है?

Write the difference between a rectangle and a rhombus.

12. किसी वर्ग की भुजा 6 सेमी है। उसका परिमाप ज्ञात करो।

Write the difference between a rectangle and a rhombus.

13. यदि किसी समान्तर चतुर्भुज के एक कोण का मान 110° है, तो अन्य कोण ज्ञात करो।

One angle of a parallelogram is 110°. Find the remaining three angles.

भाग – C : 

दीर्घ उत्तर प्रश्न (Long Answer Type)

(प्रत्येक प्रश्न 4 अंक का है)

14. किसी समचतुर्भुज की एक विकर्ण 24 सेमी और दूसरी 10 सेमी है।

(i) उसका क्षेत्रफल ज्ञात करो।

(ii) उसकी प्रत्येक भुजा की लंबाई ज्ञात करो।

The diagonals of a rhombus are 24 cm and 10 cm.

(i) Find its area.

(ii) Find the length of each side.

15. वर्ग की भुजा 10 सेमी है। उसका क्षेत्रफल और विकर्ण ज्ञात करो।

The side of a square is 10 cm. Find its area and the length of its diagonal.

16. एक समान्तर चतुर्भुज की आधार 12 सेमी और ऊँचाई 8 सेमी है। उसका क्षेत्रफल ज्ञात करो।

The base of a parallelogram is 12 cm and its height is 8 cm. Find its area.

17. एक समलंब की समान्तर भुजाएँ 8 सेमी और 12 सेमी हैं, और ऊँचाई 6 सेमी है। उसका क्षेत्रफल ज्ञात करो।

The parallel sides of a trapezium are 8 cm and 12 cm, and the height is 6 cm. 

भाग – D : 

चित्र आधारित प्रश्न 

(Geometry/Construction Based)

(प्रत्येक प्रश्न 3 अंक का है)

18. एक समान्तर चतुर्भुज बनाओ, जिसकी एक भुजा 6 सेमी और दूसरी भुजा 4 सेमी हो तथा उनमें 60° का कोण हो।

Draw a parallelogram whose one side is 6 cm, the other is 4 cm, and the included angle between them is 60°.

19. एक समलंब ABCD बनाओ जिसमें AB ∥ CD, AB = 8 सेमी, CD = 5 सेमी, और ऊँचाई = 4 सेमी हो। उसका क्षेत्रफल ज्ञात करो।

Draw a trapezium ABCD in which AB ∥ CD, AB = 8 cm, CD = 5 cm, and height = 4 cm. Find its area.

अतिरिक्त अभ्यास प्रश्न (Practice Zone – Not for Marks)

20. क्या हर आयत एक समान्तर चतुर्भुज है? कारण सहित बताओ।

Is every rectangle a parallelogram? Give reason.

21. क्या हर वर्ग एक समचतुर्भुज है? उदाहरण सहित समझाओ।

Is every square a rhombus? Explain with example.

22. दो समभुज त्रिभुजों (Equilateral Triangles) को, जिनकी प्रत्येक भुजा 4 सेमी है, जोड़कर बने चतुर्भुज (Quadrilateral) की सभी भुजाएँ और कोण ज्ञात कीजिए।

Find all the sides and the angles of the quadrilateral obtained by joining two equilateral triangles with sides 4 cm.

			🧾 Marking Scheme

Question Type	Marks per Q	No. of Qs	Total Marks
Objective (A)	1	8	8
Short Answer (B)	2	5	10
Long Answer (C)	4	4	16
Construction (D)	3	2	6
Total			40 Marks

Answer Key (with full solutions) 

Solution

01. (C) 360°

02. (C) Parallelogram

03. (A) Equal

04. (C) Not equal

05. (D) All of these

06. (C) 28 cm (Perimeter = 2×(8+6) = 28 cm)

07. (C) Trapezium

08. (C) Unequal

Section B — Short Answer Type

9. Three angles 

∠ A = 70°, 

∠ B = 80°, 

∠ C = 90°

∠ D = D

Sum of angles of a quadrilateral = 360°.

∠ A + ∠ B + ∠ C + ∠ D = 360°

70° + 80° + 90° + D = 360°

240° + D = 360°

D = 360° − 240° 

D = 120°.

10. Two properties of a parallelogram (give any two):

(i) Opposite sides are equal and parallel.

(ii) Opposite angles are equal.

(iii) Also: diagonals bisect each other.

11. Difference between a rectangle and a rhombus:

(i) Rectangle: opposite sides equal and parallel; all angles are 90°; diagonals are equal.

(ii) Rhombus: all four sides are equal; opposite sides parallel; angles need not be 90°; diagonals are perpendicular and bisect angles.

12. Side of square = 6 cm. 

Perimeter = 4 × side 

                   = 4 × 6 

                    = 24 cm.

13. One angle of a parallelogram 

∠ B = 110°, 

Sum of Adjacent angle = 180°

∠ A +  ∠ B = 180°, 

110⁰ + ∠ A = 180°, 

             ∠ A = 180° – 110°. 

             ∠ A = 70°. 

Opposite angles of a parallelogram are equal 

So ∠ A =  ∠ C = 70°, 

                  ∠ C = 70°. 

Again ∠ B =  ∠ D = 110°, 

                       ∠ D = 110°. 

Now all four angles are: 

∠ A = 70°, 

∠ B = 110°, 

∠ C = 70°

∠ D = 110°

Section C — Long Answer Type (with full working)

14. Rhombus with diagonals cm and cm.

d1 = 24 cm, 

d2 = 10 cm,

                                      1

(i) Area of rhombus = __  d1 × d2

                                       2

                                      1

                                   = __  24 × 10

                                       2

                                   = 120 cm²

(ii) Side length: 

diagonals of a rhombus are perpendicular and bisect each other. 

d1 = 24 cm, 

d2 = 10 cm,

                                             

Side = √ (d1/2)² + ( d2/2)²

                                             

Side = √ (24/2)² + ( 10/2)²

                                    

Side = √ (12)² + ( 5)²

                                

Side = √ 144 + 25

Side = √ 169

Side = 13

15. Square with side = 10 cm.

(i) Area = Side × Side

              = 10 × 10

              = 100 cm²

Diagonal = Side × √2

                = 10 × 1.732

                = 17.32

16. Parallelogram 

Base = 12 cm and 

height = 8 cm.

Area of parallelogram = Base × Height 

                                       = 12 × 8

                                       = 96cm²

17. Parallel sides in Trapezium (trapezoid) 

sides a = 8 cm and 

side b = 12cm, 

height = 6 cm.

Area of Trapezium = 1/2 (a + b) × h

                                 = 1/2 (8 + 12) × 6

                                 = 1/2 (20) × 6

                                 = 10 × 6

                                 = 60cm²

Section D — Geometry / Construction Based

18. Draw a parallelogram with sides 6 cm and 4 cm with included angle .

Steps to construct (using ruler & protractor):

1. Draw base AB = 6 cm.
2. At point A, use protractor to mark an 60⁰ angle of from AB.
3. From A along that 60⁰  ray measure and mark point D such that AD = 4 cm.
4. Through D draw a line parallel to AB  (use set-square or compass method).
5. From B draw a line parallel to AD (again with set-square/compass). 
6. The intersection of these two parallels is point C. Join BC and CD. Quadrilateral ABCD is the required parallelogram.

19. Draw trapezium ABCD with AB II CD,  AB = 8 CD = 5 cm, and height= 4cm. find its area. 

Construction steps:

1. Draw base AB = 8 cm (top longer base or bottom as you like).

2. Through point A draw a perpendicular line A and mark a point A',  at distance 4 cm

 (this will set the height). 

3. Through B draw a perpendicular line and mark point B' at distance 4 cm.

4. On the line through A' B' (which is parallel to AB), locate segment such that length CD = 5 cm. 

Position it so that its ends connect by sloping lines to A and B (Alternatively, drop perpendiculars and choose placement so that trapezium looks neat.) 

Join corresponding vertices to make trapezium.

Area calculation:

sides a = 8 cm and 

side b = 5cm, 

height = 4 cm.

Area of Trapezium = 1/2 (a + b) × h

                                 = 1/2 (8 + 5) × 4

                                 = 1/2 (13) × 4

                                 = 13 × 2

                                 = 26 cm²

20. Is every rectangle a parallelogram? Give reason.

Answer: Yes.

Reason: 

A rectangle has two pairs of opposite sides that are equal and parallel, and opposite angles equal (each angle is 90°). 

These properties satisfy the definition of a parallelogram. 

A quadrilateral whose opposite sides are parallel and equal is called IIgm.

So every rectangle is a parallelogram, but not every parallelogram is a rectangle, because a parallelogram need not have right angles.

21. Is every square a rhombus? Explain with example.

Answer: Yes.

Reason: A rhombus is a quadrilateral with all four sides equal. A square has all four sides equal and all angles 90°. 

Since a square satisfies the defining property of a rhombus (all sides equal), so every square is a rhombus.

But every rhombus is not necessarily a square, because a rhombus may not have right angles.

Example: A square of side 5 cm has all sides = 5 cm so it is a rhombus. 

But note: every rhombus is not necessarily a square, because a rhombus may not have right angles.

Q. 22 Let’s solve step-by-step carefully.

🧩 Given:

Two equilateral triangles, each having side = 4 cm.

We have to find all sides and angles of the quadrilateral obtained by joining them.

🧭 Case 1: Joined along one side (common base)

If we place two equilateral triangles △ABC and △CBD with one side BC common, then:

Let both triangles have side = 4 cm.

So:

AB = BC = AC = BD = CD = 4 cm

Now, the quadrilateral ABCD is formed by joining them along side BC.

🔹 Step 1: Identify the figure formed

When two equilateral triangles are joined along a common side, the figure obtained is a rhombus.

Reason:

All sides are equal (4 cm).

Adjacent sides are not collinear.

The diagonals bisect each other at 60° and 120°.

🔹 Step 2: Find the sides

All sides are equal to the sides of the triangles:

AB = BC = CD = DA = 4 cm

🔹 Step 3: Find the angles

Each equilateral triangle has internal angle = 60°.

When joined along one side, the angle between the adjacent triangles = 120° (supplementary to 60°).

Thus, in quadrilateral :

Sum of Adjacent angle = ∠ A + ∠ B 

                                       = 60° + 60° 

                                       = 120°

Verification 

Sum of interior angles of a quadrilateral = 360°

=> 60° + 120° + 60° + 120° = 360° ✅