Mathematics
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 figure, if AB = PQ and PQ = XY, then AB = XY.
Answer
(i) False
Reason — There are infinite no. of lines that passes through a single point.
(ii) False
Reason — By Euclid's axiom : Given two distinct points, there is a unique line that passes through them.
(iii) True
Reason — By Euclid's postulate : A terminated line can be produced indefinitely on both side.
(iv) True
Reason — If two circle are equal then there radii are equal, because if two circle are equal then on superimposing them their center and boundaries coincides and inscribe equal area thus their radii are equal.
(v) True
Reason — According to Euclid's First Axiom,"Things which are equal to the same thing are equal to one another".
Since, AB = PQ and PQ = XY
∴ AB = XY.
Related Questions
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
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.
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.