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.
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.
Solution and Explanation
There are various undefined terms in the given postulates. The given postulates are consistent because they refer to two different situations. Also, it is impossible to deduce any statement that contradicts any well known axiom and postulate.
These postulates do not follow from Euclid’s postulates. They follow from the axiom, “Given two distinct points, there is a unique line that passes through them”.
Top Questions on Euclid’s Definitions, Axioms and Postulates
- 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.
- CBSE Class IX
- Mathematics
- Euclid’s Definitions, Axioms and Postulates
- 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 Class IX
- Mathematics
- Euclid’s Definitions, Axioms and Postulates
In Fig. 5.10, if AC = BD, then prove that AB = CD.
- CBSE Class IX
- Mathematics
- Euclid’s Definitions, Axioms and Postulates
- 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.
- CBSE Class IX
- Mathematics
- Euclid’s Definitions, Axioms and Postulates
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
- CBSE Class IX
- Mathematics
- Euclid’s Definitions, Axioms and Postulates
Questions Asked in CBSE Class IX exam
- The author speaks both of people who were very aware of the differences among them and those who tried to bridge these differences. Can you identify such people in the text?
- CBSE Class IX
- My childhood
- Explain what the reasons for the following are.
(i) Einstein leaving the school in Munich for good.
(ii) Einstein wanting to study in Switzerland rather than in Munich.
(iii) Einstein seeing in Mileva an ally.
(iv) What do these tell you about Einstein?- CBSE Class IX
- A truly beautiful mind
- (i) The kind of person the doctor is (money, possessions)
(ii) The kind of person he wants to be (appearance, ambition)- CBSE Class IX
- The snake and the mirror
In Fig. 9.26, A, B, C and D are four points on a circle. AC and BD intersect at a point E such that ∠ BEC = 130° and ∠ ECD = 20°. Find ∠ BAC.
- CBSE Class IX
- Angle Subtended by an Arc of a Circle
- Name the two temples the author visited in Kathmandu.
- CBSE Class IX
- Kathmandu