Exercise-5.1

Q1. Which of the following statements are true and which are false? Give reasons for your answers.

1. Only one line can pass through a single point.
2. There are an infinite number of lines which pass through two distinct points.
3. A terminated line can be produced indefinitely on both the sides.
4. If two circles are equal, then their radii are equal.
5. In Fig. 5.9, if AB = PQ and PQ = XY, then AB = XY.

Solutions:

• (i) Only one line can pass through a single point.:
  Answer: False
  Reason: Through a single point, an infinite number of lines can pass. Hence, there is no unique line for one point
• (ii) There are an infinite number of lines which pass through two distinct points.:
  Answer: False
  Reason: According to Euclid’s Axiom, "Given two distinct points, there is a unique line that passes through them." Thus, only one line can pass through two distinct points.
• (iii) A terminated line can be produced indefinitely on both sides.:
  Answer: True
  Reason: By Euclid’s Postulate 2 — “A terminated line can be produced indefinitely.” This means a line segment can be extended on both sides endlessly.
• (iv) If two circles are equal, then their radii are equal.:
  Answer: True
  Reason: Two circles are said to be equal when their centers and circumferences coincide upon superimposition; hence, their radii are equal.
• (v) In Fig. 5.9, if AB = PQ and PQ = XY, then AB = XY:
  

  

  Answer: True
  Reason: According to Euclid’s First Axiom — “Things which are equal to the same thing are equal to one another.” Therefore, if AB = PQ and PQ = XY, then AB = XY.

Q2. 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?

1. parallel lines
2. perpendicular lines
3. line segment
4. radius of a circle
5. square

• (i) Parallel Lines
  Two straight lines lying in the same plane that do not meet each other, however far they are produced on either side, are called parallel lines. Example: Lines\[AB∥CD\] mean that line AB and line CD are parallel.
  
  Prior terms needed:
  Plane: A flat surface extending indefinitely in all directions.
  Line: A straight, infinite one-dimensional figure with length but no breadth.
  
  Definitions of prerequisite terms:
  A plane is a surface such that a straight line joining any two points on it lies entirely within it.
  A line is the shortest distance between two points and can be extended indefinitely in both directions.
• (ii) Perpendicular Lines
  Two lines lying in the same plane are said to be perpendicular if they intersect to form a right angle (an angle of 90°).
  For example, if line AB meets line CD at point O to make ∠AOC = 90°, then AB ⟂ CD.
  
  Prior terms needed:
  Plane, Line, and Right angle.
  
  Definition of a right angle:
  A right angle is formed when a straight line standing on another straight line makes the adjacent angles equal.
• (iii) Line Segment
  A line segment is a part of a straight line that is bounded by two distinct endpoints and contains all points lying between these endpoints.
  It represents the shortest distance between two fixed points on a line and has a definite measurable length.
  
  For example, if A and B are two distinct points, the part of the line joining them is the line segment AB, often denoted as\(\overline{AB}\)
  
  Prior terms that need to be defined:
  Plane: A flat surface extending indefinitely in all directions.
  Line: A straight, infinite one-dimensional figure with length but no breadth.
  
  
• (iv) Radius of a Circle
  The radius of a circle is the line segment joining the center of the circle to any point on its circumference.
  If O is the center and P is any point on the circle, the line segment OP is the radius.
  
  Prior terms needed:
  Circle, center, and circumference.
  
  Definition of a circle:
  A circle is the set of all points in a plane that are at a fixed distance (radius) from a fixed point (center).
• (v) Square
  A square is a plane figure bounded by four equal straight sides and having all four interior angles equal to right angles.
  
  Prior terms needed:
  Plane figure, side (line segment), angle, and right angle.
  
  Definitions of prerequisite terms:
  A plane figure is a flat, two-dimensional shape lying in a plane.
  A line segment is the part of a line that joins two distinct endpoints.
  An angle is the inclination between two intersecting lines measured in degrees or radians.

Q3. 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:
Undefined terms:
Yes, both postulates contain undefined terms, specifically point and line.
In Euclidean geometry, these are considered fundamental entities that are not defined using other terms but are understood intuitively:
A point has no part; it represents only a position.
A line is that which has length but no breadth and extends indefinitely in both directions.

Consistency of the postulates:
Yes, these postulates are consistent because they describe different situations that do not contradict each other.

The first postulate talks about the existence of a point C between A and B, suggesting points lying on the same line.

The second postulate asserts the existence of at least three points not on the same line, which describes a situation in which not all points are collinear.

Since one deals with collinear points and the other with non-collinear points, there is no contradiction. Hence, they are mutually consistent.

Relation to Euclid’s postulates:
These two postulates do not directly follow from Euclid’s five postulates, as none of Euclid’s original postulates explicitly mention:

• a third point lying between two given points, or
• three points that are not on the same line.

Q4. If a point \(C\) lies between two points \(A\) and \(B\) such that \(AC = BC\), then prove that \(AC = \frac{1}{2}AB\). Explain by drawing the figure

Proposition
If a point \(C\) lies between two points \(A\) and \(B\) such that \(AC = CB\), then \(AC = \frac{1}{2}AB\).
Proof
Let points \(A\), \(B\), and \(C\) be distinct and lie on the same straight line, with \(C\) between \(A\) and \(B\).

By the definition of a line segment, the total length \(AB\) is the sum of the two individual parts:

This result means that each of the two parts \(AC\) and \(CB\) is half the entire segment \(AB\)

By Euclid’s Common Notion 2 — “If equals be added to equals, the wholes are equal” — the two equal segments \(AC\) and \(CB\) together make the complete segment \(AB\)

Therefore, the equality \(AC = CB =\frac{1}{2}AB\) shows that \(C\) divides \(AB\) into two equal halves, and hence \(C\) is the midpoint of \(AB\)

Q5. 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.

Proposition
Every line segment has one and only one midpoint.

Given
A line segment \(AB\) with endpoints \(A\) and \(B\).
To Prove
There exists one and only one point \(C\) such that \(AC = CB\).

Proof
Let line segment \(AB\) as in above figure.
By Euclid’s postulate, a straight line segment can be drawn joining any two given points; this is our line segment \(AB\).
Now, let \(C\) be a point on \(AB\) such that \(AC = CB\).
Then, by definition, \(C\) is the midpoint of \(AB\).
To show uniqueness, suppose, for contradiction, that another point \(D\) on the same segment \(AB\) is also a midpoint.
Then, by the definition of a midpoint, \[AD=DB \text{ and }AC=CB\] We shall now compare these.
Since \(A\), \(C\), \(D\), and \(B\) are all on the same straight line, either \(C\) and \(D\) coincide or one of them lies between the other and one of the endpoints.
Subtracting the two equalities, \[(AC−AD)=(CB−DB)\] Now, if \(C\) and \(D\) are distinct, the difference between \(AC\) and \(AD\) equals the difference between \(CB\) and \(DB\), so both sides are equal in magnitude but opposite in direction.

Hence, \[CD=−CD\] Adding CD to both sides gives \[2CD=0\] Therefore, \[CD=0\] This means that the distance between \(C\) and \(D\) is zero, implying \(C\) and \(D\) coincide.
Thus, our supposition that there can be two midpoints leads to a contradiction.

Q6. In Fig. 5.10, if AC = BD, then prove that AB = CD.

Given
Points \(A,\; B,\; C,\; D\) are collinear in that order.
\(AC = BD\)

To Prove
\[AB = CD\] Proof On the straight line, the arrangement is: \[A — B — C — D\] Express the lengths involved, using the properties of line segments:
The length AC can be broken into: \[AC=AB+BC\] The length BD can be written as: \[BD=BC+CD\] Given that \(AC = BD\), substitute the above: \[AB+BC=BC+CD\] Applying Euclid’s Common Notion 3 (If equals are subtracted from equals, the remainders are equal), subtract BC from both sides: \[AB=CD\] Therefore, it is proved that \(AB\) is equal to \(CD\).

Q7. 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.)

Axiom 5 in Euclid’s list states:
“The whole is greater than the part.”

This statement is regarded as a universal truth because it applies not just in geometry or mathematics, but in all aspects of logical thinking and everyday experience.

• Whenever something is divided into parts, every single part is necessarily smaller than the object as a whole. For example, if a line segment is broken into sub-segments, each piece is less than the original segment.


• Similarly, if you consider any complete object—say, a book and each of its chapters, or a pizza and each slice—the total (the whole) is always larger than one piece (a part).