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:

We will use Euclid’s axioms here to check these postulates.

Yes, these postulates contain undefined terms like point and line.

These two statements are consistent as they talk about two different situations meaning different things.

These statements do not follow Euclid’s postulates but one of the axioms about “Given any two points, a unique line that passes through them” is followed.

☛ Check: NCERT Solutions Class 9 Maths Chapter 5

Video Solution:

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.

NCERT Solutions Class 9 Maths Chapter 5 Exercise 5.1 Question 3

Summary:

The two postulates "given any two distinct points A and B there exists a third point C which is in between A and B" and "there exist at least three points that are not on the same line" contain undefined terms, are consistent as they talk about two different situations, do not follow Euclid’s postulates but one of the axiom about “Given any two points, a unique line that passes through them” is followed.

☛ Related Questions:

• 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 fig. 5.9, if AB = PQ and PQ = XY, then AB = XY.
• 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 linesii) Perpendicular linesiii) Line Segmentiv) Radius of a circlev) Square
• 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.