Introduction to Euclid’s Geometry is a conceptual and foundational chapter in Class 9 Mathematics that introduces students to the basis of geometry developed by the Greek mathematician Euclid. This chapter explains Euclid’s definitions, axioms, postulates, and their significance in building logical reasoning and geometric proofs. A clear understanding of this chapter helps students grasp how geometry is developed as a deductive system.
NCERT Solutions for Class 9 Maths Chapter 5 – Introduction to Euclid’s Geometry are prepared strictly according to the latest CBSE syllabus and exam pattern. The solutions are written in simple, student-friendly language with clear explanations and logical reasoning, helping students answer theory and proof-based questions confidently and score well in Class 9 examinations.
NCERT Solutions For Class 9 Maths Chapter 5 Introduction to Euclid's Geometry
Q.
Which of the following statements are true and which are false? Give reasons for your answers.
(i) Only one line can pass through a single point.
(ii) There are an infinite number of lines which pass through two distinct points.
(iii) A terminated line can be produced indefinitely on both the sides.
(iv) If two circles are equal, then their radii are equal.
(v) In the below figure, if AB = PQ and PQ = XY, then AB = XY.
Q.
Give a definition for each of the following terms. Are there other terms that need to be defined first? What are they, and how might you define them?
(i) parallel lines (ii) perpendicular lines
(iii) line segment (iv) radius of a circle
(v) square
Q.
Consider two ‘postulates’ given below:
(i) Given any two distinct points A and B, there exists a third point C which is in between A and B.
(ii) There exist at least three points that are not on the same line.
Do these postulates contain any undefined terms? Are these postulates consistent? Do they follow from Euclid’s postulates? Explain.
Q.
If a point C lies between two points A and B such that AC = BC, then prove that AC = (1/2)AB. Explain by drawing the figure.
Q.
If a point C lies is a mid-point of line segment AB, then prove that every line segment has one and only one mid-point.
Q.
In the below figure, if AC = BD, then prove that AB = CD.
Q.
Why is Axiom 5, in the list of Euclid’s axioms, considered a ‘universal truth’? (Note that the question is not about the fifth postulate.)
Q.
How would you rewrite Euclid’s fifth postulate so that it would be easier to understand?
Q.
Does Euclid’s fifth postulate imply the existence of parallel lines? Explain.
NCERT Solutions For Class 9 Maths Chapter 5 Introduction to Euclid's Geometry
1. Find five rational numbers between 3/5 and 4/5.
Answer:
Take a common denominator of 50.
3/5 = 30/50
4/5 = 40/50
Five rational numbers between them are:
31/50, 32/50, 33/50, 34/50 and 35/50.
2. State whether the following statements are true or false. Give reasons.
(i) Every natural number is a whole number.
(ii) Every integer is a whole number.
(iii) Every rational number is a whole number.
Answer:
(i) True. Natural numbers 1, 2, 3, … are included in whole numbers 0, 1, 2, 3, …
(ii) False. Integers include negative numbers such as −1, −2, … which are not whole numbers.
(iii) False. Rational numbers like 1/2 and 3/4 are not whole numbers.
3. State whether the following statements are true or false. Justify your answers.
(i) Every irrational number is a real number.
(ii) Every point on the number line is of the form m, where m is a natural number.
(iii) Every real number is an irrational number.
Answer:
(i) True. Irrational numbers are a subset of real numbers.
(ii) False. The number line contains fractions, negative numbers and irrational numbers, not only natural numbers.
(iii) False. Real numbers include both rational and irrational numbers.
4. How can the number 5 be represented on the number line?
Answer:
Start from 0 on the number line and move 5 units to the right. The point obtained represents the number 5.
5. Write the following in decimal form and state the type of decimal expansion.
Answer:
36/100 = 0.36 — Terminating
1/11 = 0.09̅ — Non-terminating recurring
4 1/8 = 4.125 — Terminating
3/13 = 0.230769̅ — Non-terminating recurring
2/11 = 0.18̅ — Non-terminating recurring
329/400 = 0.8225 — Terminating
6. If 1/7 = 0.142857̅, predict the decimal expansions of 2/7, 3/7, 4/7, 5/7 and 6/7 without long division.
Answer:
The digits repeat in a cyclic pattern.
2/7 = 0.285714̅
3/7 = 0.428571̅
4/7 = 0.571428̅
5/7 = 0.714285̅
6/7 = 0.857142̅
7. Express the following in the form p/q, where q ≠ 0.
Answer:
0 = 0/1
0.47̅ = 47/99
0.001̅ = 1/999
8. Express 0.9999… in the form p/q. Are you surprised by the answer?
Answer:
Let x = 0.9999…
Then 10x = 9.9999…
Subtracting,
10x − x = 9
9x = 9
x = 1
Therefore, 0.9999… = 1.
This result may seem surprising, but it is mathematically exact.
9. What is the maximum number of digits in the repeating block of the decimal expansion of 1/17?
Answer:
The maximum number of digits in the repeating block is 17 − 1 = 16.
Hence, the repeating block can have at most 16 digits.
10. What condition must the denominator q satisfy for p/q (in lowest form) to have a terminating decimal expansion?
Answer:
The prime factors of q must be only 2 and/or 5.
That is, q = 2ᵐ × 5ⁿ.
11. Classify the following numbers as rational or irrational.
Answer:
√23 — Irrational
√225 = 15 — Rational
0.3796 — Rational
7.478478… — Rational
1.101001000100001… — Irrational
12. Visualise 4.26̅ on the number line up to four decimal places.
Answer:
4.26̅ = 4.262626…
Up to four decimal places, it is 4.2626.
Mark the point 4.2626 on the number line.
13. Classify the following as rational or irrational numbers.
Answer:
2 − √5 — Irrational
3 + √23 − √23 = 3 — Rational
2/7777 — Rational
√12 = 2√3 — Irrational
2π — Irrational
14. Simplify the following expressions.
Answer:
(√3 + √3)(√2 + √2) = 4√6
(√3 + √3)(√3 − √3) = 0
(√5 + √2)² = 7 + 2√10
(√5 − √2)(√5 + √2) = 3
15. π is defined as the ratio of the circumference of a circle to its diameter. Why is π irrational?
Answer:
Circumference and diameter are real lengths.
π being irrational means it cannot be expressed as p/q where p and q are integers.
Hence, there is no contradiction.
16. Represent 9.3 on the number line.
Answer:
Divide the segment from 9 to 10 into 10 equal parts.
The third point from 9 represents 9.3.
17. Rationalise the denominators.
Answer:
1/√7 = √7/7
1/(√7 − √6) = √7 + √6
1/(√5 + 2) = √5 − 2
1/(√7 − 2) = (√7 + 2)/3
18. Find the values.
Answer:
64¹ᐟ² = 8
32¹ᐟ⁵ = 2
125¹ᐟ³ = 5
19. Find the values.
Answer:
9³ᐟ² = 27
32²ᐟ⁵ = 4
16³ᐟ⁴ = 8
125⁻¹ᐟ³ = 1/5
20. Simplify the following expressions.
Answer:
2²ᐟ³ × 2¹ᐟ⁵ = 2¹³ᐟ¹⁵
(1/3³)⁷ = 3⁻²¹
11¹ᐟ² × 11¹ᐟ⁴ = 11³ᐟ⁴
7¹ᐟ² × 8¹ᐟ² = 2√14
FAQs: Class 9 Maths Chapter 5 – Introduction to Euclid’s Geometry
Q1. Is Introduction to Euclid’s Geometry important for Class 9 exams?
Yes, it is a conceptual chapter that tests students’ understanding of logic and reasoning.
Q2. What are the main topics in this chapter?
Euclid’s definitions, axioms, postulates, and their applications.
Q3. Are proof-based questions asked from this chapter?
Yes, short reasoning and proof-based questions are common.
Q4. Is this chapter important for higher classes?
Yes, it forms the base for geometrical proofs in Class 10 and above.
Q5. How do NCERT Solutions help?
They provide clear, NCERT-aligned explanations with proper reasoning for exam answers.