Find the best tutors and institutes for Class 10 Tuition
Q1(i):
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.
Solution :
Step 1: Evaluation of the Proposition
The statement provided is: "Only one line can pass through a single point." To evaluate the truth value of this statement, we must analyze the spatial geometry of a point within a plane.
Step 2: Theoretical Geometric Analysis
A point, as defined by Euclid in his first definition, is "that which has no part" [Euclid's Elements, Definition 1]. In Euclidean geometry, a line is defined as "breadthless length" [Euclid's Elements, Definition 2].
If we denote a point as $P$ in a two-dimensional Euclidean plane $\mathbb{R}^2$, we consider the set of all lines $L$ that satisfy the condition $P \in L$. By the properties of incidence geometry, specifically the axioms regarding point-line incidence, an infinite number of distinct straight lines can be drawn such that they intersect at a single common point $P$.
Step 3: Visual Representation
Step 4: Logical Conclusion
- Axiomatic Contradiction: Euclid's Axioms do not limit the number of lines through a single point. Conversely, Euclid’s Postulate 1 states: "To draw a straight line from any point to any point." This implies that while a unique line exists between two distinct points, there is no restriction on the number of lines passing through a single point.
- Geometric Reality: In a plane, for any point $(x_0, y_0)$, the general equation of a line passing through it is given by $y - y_0 = m(x - x_0)$. Since the slope $m$ can take any real value ($m \in \mathbb{R}$) or be undefined (vertical line), there exists an infinite set of values for $m$, and consequently, an infinite number of lines.
Final Solution: The statement is False. An infinite number of lines can pass through a single point.
More Questions from Class 9 Mathematics Introduction to Euclid’s Geometry EXERCISE 5.1
- Q1(ii): Which of the following statements are true and which are false? Give reasons for your answers. (ii) There are an infinite number of lines which pass through two distinct points.
- Q1(iii): Which of the following statements are true and which are false? Give reasons for your answers. (iii) A terminated line can be produced indefinitely on both the sides.
- Q1(iv): Which of the following statements are true and which are false? Give reasons for your answers. (iv) If two circles are equal, then their radii are equal.
- Q1(v): Which of the following statements are true and which are false? Give reasons for your answers. (v) In Fig. 5.9, if $AB = PQ$ and $PQ = XY$, then $AB = X
- Q2(i): 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
- Q2(ii): 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? (ii) perpendicular lines
- Q2(iii): 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? (iii) line segment
- Q2(iv): 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? (iv) radius of a circle
- Q2(v): 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? (v) square
- Q3: 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.
- Q4: If a point $C$ lies between two points $A$ and $B$ such that $AC = BC$, then prove that $AC = \frac{1}{2}AB$. Explain by drawing the figure.
- Q5: In Question 4, point $C$ is called a mid-point of line segment $AB$. Prove that every line segment has one and only one mid-point.
- Q6: In Fig. 5.10, if $AC = BD$, then prove that $AB = CD$.
- Q7: 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.)
CBSE Solutions for Class 9 Mathematics Introduction to Euclid’s Geometry
Chapters in CBSE - Class 9 Mathematics
Top Tutors who teach Introduction to Euclid’s Geometry
Top Tutor
Anna Nagar Western Extn, Chennai
7 yrs of
Exp
3 students
48 hrs of classes completed
500
per hour
Student's Home
Tutor's home
I am an experienced, qualified teacher and tutor with over 6 years of experience in teaching Maths across different boards including CBSE, ICSE, IGCSE and State Board. Passionate about solving mathematical problems. over the years I have helped hundred of students overcome their fear of Maths. I am a post graduates in mathematics and under graduates in education. We are available on both home tuition and online We have lot of success stories.
Super Tutor
Badarpur, Delhi
12 yrs of
Exp
156 students
19036 hrs of classes completed
200
per hour
Online Classes
Tutor's home
Akshay is an excellent teacher. He has taught my son in 10 th standard science , with which he understood the concepts clearly and he also gave so many past question papers for my son to work out, all these made my son score 95 in board exams . I would definitely recommend Akshay for students who want to improve and get good scores in science .
Top Tutor
BRS Nagar, Ludhiana
27 hrs of classes completed
Online Classes
Student's Home
Tutor's home
I was a student in one of the top colleges of Ludhiana. I have proficiency in English . I can teach all subjects to primary wing .
Top Tutor
Electronic City Electronics City Phase 1, Bangalore
15 yrs of
Exp
5 students
78 hrs of classes completed
1000
per hour
Online Classes
Ph.D enrolled, M.Tech(Electrical eng). Over 15 years of experience in teaching at different Engineering colleges and tutoring across different boards including CBSE, ICSE, IGCSE, IBDP, Level AS/A in Mathematics & Science. Approachable to students & succeeded to boost their fundamental knowledge.
I used to struggle a lot with mathematics, but my tuition teacher has helped me improve tremendously. She explains concepts clearly and patiently, making even difficult topics easier to understand. Instead of simply giving answers, she encourages me to think independently and solve problems on my own, which has greatly improved my confidence and problem-solving skills. She provides plenty of practice and ensures that I fully understand each concept before moving on. Lessons with her are always engaging, enjoyable, and easy to follow. Most importantly, she understands my learning style and supports me whenever I face challenges. I am very grateful for her guidance and dedication.
Top Tutor
Civil Lines, Prayagraj
6 yrs of
Exp
3 students
232 hrs of classes completed
300
per hour
Online Classes
Student's Home
Tutor's home
I am a tutor with 6 years experience, academically b.Sc. , b.Ed. , m.A. & ctet qualified. I love to solve problems of students and help them in getting calm and cool. I welcome you with your problems on this account. Wish you a very very nice learning journey.
Find more Tutor for Introduction to Euclid’s Geometry in your City
- Bangalore Mathematics Tutors
- Delhi Mathematics Tutors
- Chennai Mathematics Tutors
- Gurgaon Mathematics Tutors
- Noida Mathematics Tutors
- Hyderabad Mathematics Tutors
- Mumbai Mathematics Tutors
- Chandigarh Mathematics Tutors
- Pune Mathematics Tutors
- Ghaziabad Mathematics Tutors
- Jaipur Mathematics Tutors
- Surat Mathematics Tutors
Download free CBSE - Class 9 Mathematics Introduction to Euclid’s Geometry EXERCISE 5.1 worksheets
Download Now
