INTRODUCTION TO EUCLID'S GEOMETRY - Q&A
Exercise 5.1
1. 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.
False. Infinite number of lines can pass through a single point.
Reason: Consider a point P. We can draw lines in all directions passing through P.
(ii) There are an infinite number of lines which pass through two distinct points.
False. Only one unique line can pass through two distinct points.
Reason: This is a fundamental postulate (Euclid's Postulate 1) which states that a straight line may be drawn from any one point to any other point.
(iii) A terminated line can be produced indefinitely on both the sides.
True. A terminated line (line segment) can be extended on both sides to form a line.
Reason: This corresponds to Euclid's Postulate 2.
(iv) If two circles are equal, then their radii are equal.
True. If two circles are equal, it means they are congruent and can be superimposed on each other exactly. Therefore, their centres and boundaries will coincide, meaning their radii must be equal.
(v) In Fig. 5.9, if AB = PQ and PQ = XY, then AB = XY.
True. According to Euclid's Axiom 1: "Things which are equal to the same thing are equal to one another."
Here, AB and XY are both equal to PQ, so AB must be equal to XY.
2. 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
Parallel lines are two straight lines in a plane that do not intersect at any point, no matter how far they are extended.
[Image of parallel lines]
Terms to be defined first: Plane, Point, Line.
- Point: A dot with no dimensions.
- Line: A collection of points extending indefinitely in both directions.
(ii) Perpendicular lines
Two lines are said to be perpendicular if they intersect each other at a right angle (90 degrees).
[Image of perpendicular lines]
Terms to be defined first: Line, Angle, Right Angle.
- Angle: A figure formed by two rays meeting at a common end point.
- Right Angle: An angle measuring 90 degrees.
(iii) Line Segment
A line segment is a part of a line with two distinct end points.
Terms to be defined first: Line, Point.
- Point: An exact location in space.
(iv) Radius of a circle
The radius of a circle is the line segment connecting the centre of the circle to any point on its boundary (circumference).
[Image of radius of a circle]
Terms to be defined first: Circle, Centre, Point.
- Circle: The collection of all points in a plane that are at a fixed distance from a fixed point.
(v) Square
A square is a quadrilateral (a four-sided polygon) in which all four sides are equal in length and all four angles are right angles.
Terms to be defined first: Quadrilateral, Side, Angle, Right Angle.
3. 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.
Yes, these postulates contain undefined terms like 'point' and 'line'.
Yes, these postulates are consistent because they refer to two different situations and do not contradict each other.
- The first postulate says that between two points, there are infinite points.
- The second postulate deals with points that are not collinear (not on the same line).
No, they do not follow directly from Euclid's postulates. However, they follow from Euclid's Axioms. For example, the first one assumes that a line is continuous.
4. 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.
Given: AC = BC
To prove: AC = 1/2 AB
Proof:
From the figure (draw a line segment AB with C in the middle), we know that AB coincides with AC + BC.
So, AB = AC + BC
Since AC = BC (Given), we can substitute BC with AC.
AB = AC + AC
AB = 2AC
Dividing both sides by 2:
AC = 1/2 AB.
(This uses Euclid's Axiom: If equals are added to equals, the wholes are equal.)
5. 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.
Let's assume that there are two mid-points, C and D, for the line segment AB.
Since C is a mid-point, AC = 1/2 AB ... (1)
Since D is a mid-point, AD = 1/2 AB ... (2)
From (1) and (2), we get:
AC = AD
This means point C and point D coincide (lie on top of each other).
Therefore, a line segment has one and only one unique mid-point.
6. In Fig. 5.10, if AC = BD, then prove that AB = CD.
Given: AC = BD
From the figure, we can see:
AC = AB + BC ... (1)
BD = BC + CD ... (2)
Since AC = BD (given), we can equate (1) and (2):
AB + BC = BC + CD
Subtracting BC from both sides (Euclid's Axiom: If equals are subtracted from equals, the remainders are equal):
AB = CD.
Hence proved.
7. 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.)
Euclid's Axiom 5 states: "The whole is greater than the part."
This is considered a universal truth because it holds true for everything in the universe, not just in mathematics. Whether it is a quantity, a physical object, or a region of space, a part is always contained within the whole, making the whole magnitude larger than the constituent part. This is fundamentally true by the definition of 'whole' and 'part'.