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 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.
(i) False. This can be seen visually by the student.
(ii) False. This contradicts Axiom 5.1.
(iii) True. (Postulate 2)
(iv) True. If you superimpose the region bounded by one circle on the other, then they coincide. So, their centres and boundaries coincide. Therefore, their radii coincide.
(v) True. The first axiom of Euclid.
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
There are several undefined terms which the student should list. They are consistent, because they deal with two different situations — (i) says that given two points A and B, there is a point C lying on the line in between them; (ii) says that given A and B, you can take C not lying on the line through A and B.
These ‘postulates’ do not follow from Euclid’s postulates. However, they follow from Axiom 5.1.
Consider two ‘postulates’:
(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.
There are several undefined terms which the student should list. They are consistent because they deal with two different situations — (i) says that given two points A and B, there is a point C lying on the line between them; (ii) says that given A and B, you can take C not lying on the line through A and B.
These ‘postulates’ do not follow from Euclid’s postulates. However, they follow from Axiom 5.1.
If a point C lies between two points A and B such that AC = BC, then prove that AC = (1/2) AB.
AC = BC
So, AC + AC = BC + AC (Equals are added to equals)
i.e., 2AC = AB (BC + AC coincides with AB)
Therefore, AC = 1/2 AB
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.
Make a temporary assumption that different points C and D are two mid-points of AB. Now, you show that points C and D are not two different points.
In Fig. 5.10, if AC = BD, then prove that AB = CD.
AC = BD (Given)
AC = AB + BC (Point B lies between A and C)
BD = BC + CD (Point C lies between B and D)
Substituting, AB + BC = BC + CD
So, AB = CD (Subtracting equals from equals)
Why is Axiom 5, in the list of Euclid’s axioms, considered a ‘universal truth’?
Since this is true for anything in any part of the world, this is a universal truth.