1Which 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.
Answer & Reference:
(i) False. Infinitely many lines can pass through a single point. (Based on Postulate 1, page 64)
(ii) False. According to Axiom 5.1 (page 64), "Given two distinct points, there is a unique line that passes through them."
(iii) True. This is exactly what Euclid's Postulate 2 states: "A terminated line can be produced indefinitely." (Page 64)
(iv) True. If two circles are equal, they coincide with each other. By Euclid's Axiom (4), "Things which coincide with one another are equal to one another." Therefore, their radii must be equal.
(v) True. By Euclid's Axiom (1), "Things which are equal to the same thing are equal to one another." Since AB = PQ and PQ = XY, then AB = XY. (Page 63)
2Give 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
Answer & Reference:
To define the given terms, we need to define some basic terms first:
- Point: That which has no part (Euclid's Definition 1, page 62)
- Line: Breadthless length (Euclid's Definition 2, page 62)
- Straight line: A line which lies evenly with the points on itself (Euclid's Definition 4, page 62)
- Surface: That which has length and breadth only (Euclid's Definition 5, page 62)
- Plane surface: A surface which lies evenly with the straight lines on itself (Euclid's Definition 7, page 62)
- Right angle: When a straight line set up on a straight line makes the adjacent angles equal to one another, each of the equal angles is right (implied in Postulate 4, page 64)
Now the definitions:
(i) Parallel lines: Two lines in a plane that never meet, even when extended indefinitely in both directions. They maintain a constant distance between them.
(ii) Perpendicular lines: Two lines that intersect at a right angle (90°).
(iii) Line segment: A part of a line with two endpoints. (Euclid called this a "terminated line" as in Postulate 2, page 64)
(iv) Radius of a circle: The distance from the center of a circle to any point on its circumference. (Implied in Postulate 3, page 64)
(v) Square: A quadrilateral with all sides equal and all angles right angles. (Referenced in the context of altars in page 61)
3Consider 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.
Answer & Reference:
- Undefined terms: Yes, these postulates contain undefined terms like "point", "line", and "between". In Euclid's geometry, point, line, and plane are considered undefined terms (page 63).
- Consistency: Yes, these postulates are consistent because they deal with different situations and do not contradict each other or any of Euclid's axioms/postulates.
- Relation to Euclid's postulates: These postulates do not directly follow from Euclid's postulates. However, they are consistent with them. Postulate (ii) is related to the idea that not all points are collinear, which is necessary for defining a plane. Postulate (i) introduces the concept of "betweenness" which is not explicitly covered in Euclid's original postulates but is necessary for modern geometry. (Based on discussion of undefined terms on page 63)
4If a point C lies between two points A and B such that AC = BC, then prove that AC = ½ AB. Explain by drawing the figure.
Answer & Reference:
Given: Point C lies between A and B such that AC = BC.
To prove: AC = ½ AB
Fig. 5.7: Point C between A and B
Proof:
Since C lies between A and B,
AB = AC + BC (By Example 1, page 65)
But AC = BC (Given)
Therefore, AB = AC + AC = 2AC
Hence, AC = ½ AB
(Based on Example 1, page 65)
5In 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.
Answer & Reference:
Let us assume that a line segment AB has two mid-points C and D.
Assuming two mid-points C and D
If C is the mid-point, then AC = CB = ½ AB
If D is the mid-point, then AD = DB = ½ AB
From the figure:
AC + CD = AD
Substituting values:
½ AB + CD = ½ AB
This implies CD = 0
Therefore, C and D coincide.
Hence, every line segment has one and only one mid-point.
(Based on the concept from Question 4, page 68)
6In Fig. 5.10, if AC = BD, then prove that AB = CD.
Answer & Reference:
Given: AC = BD
To prove: AB = CD
Fig. 5.10: Points A, B, C, D on a line
From the figure:
AC = AB + BC
BD = BC + CD
Given AC = BD,
Therefore, AB + BC = BC + CD
Subtracting BC from both sides (using Euclid's Axiom 3: "If equals are subtracted from equals, the remainders are equal"),
AB = CD
(Based on Euclid's Axiom 3, page 63)
7Why is Axiom 5, in the list of Euclid's axioms, considered a 'universal truth'? (Note that the question is not about the fifth postulate.)
Answer & Reference:
Axiom 5 states: "The whole is greater than the part."
This is considered a universal truth because it applies to everything in the universe, not just geometric figures. For example:
- In numbers: 5 > 3 (5 is the whole, 3 is a part)
- In sets: The set of natural numbers is greater than the set of even numbers (though both are infinite, the former contains the latter)
- In physical objects: A book is greater than a single page of that book
This axiom is not specific to geometry but is a fundamental truth in all areas of mathematics and life. That's why it's considered a "universal truth".
(Based on discussion of Axiom 5, page 63)
SSummary of Key Concepts
Euclid's Definitions (Book 1):
Definition
Statement
Page
1
A point is that which has no part.
62
2
A line is breadthless length.
62
3
The ends of a line are points.
62
4
A straight line is a line which lies evenly with the points on itself.
62
5
A surface is that which has length and breadth only.
62
6
The edges of a surface are lines.
62
7
A plane surface is a surface which lies evenly with the straight lines on itself.
62
Euclid's Axioms (Common Notions):
Things which are equal to the same thing are equal to one another. (Page 63)
If equals are added to equals, the wholes are equal. (Page 63)
If equals are subtracted from equals, the remainders are equal. (Page 63)
Things which coincide with one another are equal to one another. (Page 63)
The whole is greater than the part. (Page 63)
Things which are double of the same things are equal to one another. (Page 63)
Things which are halves of the same things are equal to one another. (Page 63)
Euclid's Postulates:
A straight line may be drawn from any one point to any other point. (Page 64)
A terminated line can be produced indefinitely. (Page 64)
A circle can be drawn with any centre and any radius. (Page 64)
All right angles are equal to one another. (Page 64)
If a straight line falling on two straight lines makes the interior angles on the same side of it taken together less than two right angles, then the two straight lines, if produced indefinitely, meet on that side on which the sum of angles is less than two right angles. (Page 64-65)