Class 9 Mathematics Exercise-5.1 NCERT Solutions Olympiad Board Exam

Chapter 5

INTRODUCTION TO EUCLID’S GEOMETRY

Step-by-step NCERT solutions with stress–strain analysis and exam-oriented hints for Boards, JEE & NEET.

7 Questions

15–25 min Ideal time

Q1 Now at

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NUMERIC2 marks

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

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

📘 Concept & Theory Concept Used in this Exercise ›

This exercise is based on the basic ideas introduced by the Greek mathematician Euclid. The questions mainly use Euclid’s definitions, axioms and postulates.

Concept	Explanation
Point	A point indicates an exact position. It has no length, breadth or thickness.
Line	A line extends endlessly in both directions.
Line Segment	A line segment has two fixed end points.
Euclid’s First Axiom	Things which are equal to the same thing are equal to one another.
Euclid’s Second Postulate	A terminated line can be produced indefinitely.
Unique Line Property	Through two distinct points, only one unique line can pass.

🗺️ Solution Roadmap Step-by-step Plan ›

Think about all possible directions through one point.
Check whether only one line is possible.
Use geometric understanding of lines through a point.

📊 Graph / Figure Graph / Figure ›

✏️ Solution Part (i) ›

Step-by-step Solution · 5 steps

The statement says that only one line can pass through a single point.
Consider a point \(P\).
Through this point, we can draw lines in infinitely many directions.
Therefore, infinitely many lines can pass through one point.
Hence, the given statement is False.

📝 Part(ii) ›

There are an infinite number of lines which pass through two distinct points.

🗺️ Solution Roadmap Step-by-step Plan ›

Recall the unique line property.
Take two distinct points.
Check how many lines can join them.

📊 Graph / Figure Graph / Figure ›

✏️ Solution Part (ii) ›

Step-by-step Solution · 5 steps

Let \(A\) and \(B\) be two distinct points.
According to Euclidean Geometry:
“Through two distinct points, only one unique line can pass.”
Therefore, it is not possible to draw infinitely many lines through the same two distinct points.
Hence, the given statement is False.

📝 Part(iii) ›

A terminated line can be produced indefinitely on both the sides.

🗺️ Solution Roadmap Step-by-step Plan ›

Recall Euclid’s Second Postulate.
Understand meaning of terminated line.
Check whether extension is possible.

📊 Graph / Figure Graph / Figure ›

✏️ Solution Part(iii) ›

Step-by-step Solution · 5 steps

A terminated line means a line segment having fixed end points.
Euclid’s Second Postulate states:
"A terminated line can be produced indefinitely.”
This means the line segment can be extended endlessly in both directions.
Therefore, the given statement is True.

📝 Part(iv) ›

If two circles are equal, then their radii are equal.

🗺️ Solution Roadmap Step-by-step Plan ›

Recall definition of equal circles.
Understand relation between equal circles and radius.

✏️ Solution Complete Solution ›

Step-by-step Solution · 3 steps

Two circles are said to be equal if their radii are equal.
Therefore, if two circles are equal, then their radii must also be equal.
Hence, the given statement is True.

📝 Part(v) ›

In Fig. 5.9, if AB = PQ and PQ = XY, then AB = XY.

📊 Graph / Figure Graph / Figure ›

🗺️ Solution Roadmap Step-by-step Plan ›

Observe equality relation among line segments.
Use Euclid’s First Axiom.
Conclude equality between AB and XY.

✏️ Solution Part (v) ›

Step-by-step Solution · 9 steps

It is given that:
\[AB = PQ\]
Also,
\[PQ = XY\]
According to Euclid’s First Axiom:
“Things which are equal to the same thing are equal to one another.”
Since both \(AB\) and \(XY\) are equal to \(PQ\),
\[AB = XY\]
Therefore, the given statement is True.

🎯 Exam Significance Exam Significance ›

Questions based on Euclid’s axioms and postulates are frequently asked in CBSE school examinations.
Understanding logical reasoning in geometry helps in Olympiads and NTSE-type examinations.
Euclid Geometry develops proof-writing skills which are essential in higher mathematics.
Basic geometric logic is useful for competitive examinations involving quantitative aptitude and reasoning.

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Q2 →

📘 Concept & Theory Concept Used in this Exercise ›

In this exercise, we study the basic geometrical definitions used in Euclid’s Geometry. Geometry is built step-by-step using simple undefined terms such as point, line and plane. More complicated figures are defined using these basic terms.

Basic Term	Meaning
Point	An exact position without length, breadth or thickness.
Line	A straight path extending endlessly in both directions.
Plane	A flat surface extending indefinitely in all directions.
Angle	The figure formed when two rays meet at a common point.
Circle	A set of all points in a plane at a fixed distance from a fixed point.

Solution Strategy

First identify the geometrical object being defined.
Recall the basic properties of the figure.
Mention important prerequisite terms needed in the definition.
Define those prerequisite terms clearly.
Use proper mathematical language and standard geometry terminology.

📝 Part (i) ›

parallel lines

🗺️ Solution Roadmap Step-by-step Plan ›

Understand what happens when two lines never meet.
Check whether the lines lie in the same plane.
Write prerequisite terms like line and plane.

📊 Graph / Figure Graph / Figure ›

✏️ Solution Part (i) ›

Step-by-step Solution · 8 steps

Two straight lines lying in the same plane which do not intersect each other, even if produced indefinitely on both sides, are called parallel lines.
Example
\[AB \parallel CD\]
Terms Needed Before Definition
- Plane
- Line
Definitions of Prerequisite Terms
Plane: A flat surface extending indefinitely in all directions.
Line: A straight path extending endlessly in both directions.

📝 Part (ii) ›

Perpendicular Lines

🗺️ Solution Roadmap Step-by-step Plan ›

Observe intersecting lines.
Check whether the angle formed is a right angle.
Define right angle before perpendicular lines.

📊 Graph / Figure Graph / Figure ›

✏️ Solution Part (ii) ›

Step-by-step Solution · 7 steps

Two lines in the same plane are said to be perpendicular if they intersect each other to form a right angle.
Example
\[AB \perp CD\]
Terms Needed Before Definition
- Line
- Plane
- Right angle
Definitions of Prerequisite Terms
Right Angle: An angle measuring \[ 90^\circ \] is called a right angle.

📝 Part (iii) ›

Line Segment

🗺️ Solution Roadmap Step-by-step Plan ›

Start from the idea of a line.
Identify fixed end points.
Explain measurable length.

📊 Graph / Figure Graph / Figure ›

✏️ Solution Part (iii) ›

Step-by-step Solution · 9 steps

A line segment is a part of a line bounded by two fixed end points.
It has a definite measurable length.
Example
\[\overline{AB}\]
Terms Needed Before Definition
- Point
- Line
Definitions of Prerequisite Terms
Point: An exact position without dimensions.
Line: A straight path extending endlessly in both directions.

📝 Part (iv) ›

Radius of a Circle

🗺️ Solution Roadmap Step-by-step Plan ›

Identify the centre of the circle.
Join the centre to a point on the circle.
Define related terms like circle and circumference.

📊 Graph / Figure Graph / Figure ›

✏️ Solution Part (iv) ›

Step-by-step Solution · 8 steps

The radius of a circle is the line segment joining the centre of the circle to any point on the circle.
If \(O\) is the centre and \(P\) is a point on the circle, then
\[OP\]
is the radius.
Terms Needed Before Definition
- Circle
- Centre
Definitions of Prerequisite Terms
Circle: A set of all points in a plane at a fixed distance from a fixed point.

📝 Part (v) ›

Square

🗺️ Solution Roadmap Step-by-step Plan ›

Observe sides of the figure.
Check equality of all sides.
Check all interior angles.

📊 Graph / Figure Graph / Figure ›

✏️ Solution Part (v) ›

Step-by-step Solution · 7 steps

A square is a quadrilateral having:
- all four sides equal, and
- all four interior angles equal to right angles.
Terms Needed Before Definition
- Quadrilateral
- Line segment
- Right angle
Definitions of Prerequisite Terms
Quadrilateral: A closed plane figure formed by four line segments.
Right Angle: An angle measuring \[ 90^\circ \] is called a right angle.

🎯 Exam Significance Exam Significance ›

Definitions from Euclid Geometry are frequently asked in school examinations.
Understanding geometric terminology helps in proof-based questions.
Competitive examinations often test conceptual clarity of geometrical figures.
Proper understanding of definitions builds the foundation for higher geometry.

← Q1

2 / 7 · 29%

Q3 →

NUMERIC3 marks

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.

📘 Concept & Theory Concept Used in this Question ›

This question is based on the ideas of undefined terms, consistency of postulates, and Euclid’s geometric system. In Euclidean Geometry, some basic objects are accepted without formal definitions. These undefined terms are then used to define all other geometrical concepts.

Concept	Explanation
Undefined Terms	Basic geometrical objects accepted without formal definition, such as point, line and plane.
Postulate	A mathematical statement accepted as true without proof.
Consistency	Two statements are consistent if they do not contradict each other.
Collinear Points	Points lying on the same straight line.
Non-Collinear Points	Points that do not lie on the same straight line.

How to Solve Questions Based on Postulates

Read each postulate carefully and identify important mathematical terms.
Check whether the terms are defined or undefined in Euclidean Geometry.
Examine whether the statements contradict each other.
Compare the statements with Euclid’s original postulates.
Write conclusions with proper geometrical reasoning.

📊 Graph / Figure Graph / Figure ›

Postulate (i)

Postulate (ii)

Visual Understanding of the Postulates

🗺️ Solution Roadmap Step-by-step Plan ›

Identify undefined terms appearing in the postulates.
Check whether the two statements contradict each other.
Compare these statements with Euclid’s original postulates.
Draw a logical conclusion with proper explanation.

✏️ Solution Complete Solution ›

Step-by-step Solution · 25 steps

Identify Undefined Terms
Yes, these postulates contain undefined terms.
The undefined terms used are:
- Point
- Line
In Euclidean Geometry, these terms are accepted without formal definitions.
A point represents only a position and has no dimensions.
A line is a straight path extending indefinitely in both directions.
Check Consistency of the Postulates
The two postulates are consistent because they do not contradict each other.
In the first postulate, points \(A\), \(C\), and \(B\) lie on the same line, with point \(C\) between \(A\) and \(B\).
\[A \;-\; C \;-\; B\]
In the second postulate, there exist at least three points which are not on the same line.
This means the points are non-collinear.
Since one postulate discusses collinear points and the other discusses non-collinear points, there is no contradiction.
Therefore, the postulates are consistent.
Relation with Euclid’s Postulates
These postulates do not directly follow from Euclid’s five postulates.
Euclid’s first postulate states:
“A straight line may be drawn from any one point to any other point.”
This supports the idea that points can lie on the same line.
However, Euclid’s postulates do not explicitly state:
- that a third point lies between two given points, or
- that there exist three non-collinear points.
Therefore, these statements are compatible with Euclid’s geometry, but they are not directly derived from Euclid’s postulates.
Final Conclusion
- The undefined terms are point and line.
- The postulates are consistent because they do not contradict each other.
- These postulates do not directly follow from Euclid’s postulates, though they are compatible with Euclidean Geometry.

🎯 Exam Significance Exam Significance ›

Questions on axioms, postulates and undefined terms are frequently asked in CBSE examinations.
Understanding consistency of statements improves logical reasoning ability.
These concepts form the foundation of proof-based geometry in higher classes.
Competitive examinations often test conceptual understanding rather than direct formulas.

← Q2

3 / 7 · 43%

Q4 →

📘 Concept & Theory Concept Used in this Question ›

This question is based on the addition of line segments and the concept of midpoint. When a point lies between two points on a line, the whole line segment is equal to the sum of its parts.

Concept	Explanation
Collinear Points	Points lying on the same straight line.
Line Segment Addition	If a point lies between two points on a line, then the total segment equals the sum of smaller segments.
Equal Segments	Two segments having the same length are called equal segments.
Midpoint	A point dividing a line segment into two equal parts is called midpoint.
Euclid’s Common Notion 2	If equals are added to equals, then the wholes are equal.

Solution Strategy

Draw the figure showing point \(C\) between \(A\) and \(B\).
Use the segment addition property: \[ AB = AC + CB \]
Substitute the given condition: \[ AC = CB \]
Simplify the equation step-by-step.
Divide both sides by \(2\) to obtain the required result.

📊 Graph / Figure Graph / Figure ›

Point \(C\) lies between points \(A\) and \(B\), dividing the line segment into two equal parts.

✏️ Solution Complete Solution ›

Step-by-step Solution · 24 steps

Given Information
Point \(C\) lies between points \(A\) and \(B\).
Also,
\[AC = CB\]
To Prove
\[AC = \frac{1}{2}AB\]
Proof
Use Segment Addition Property
Since point \(C\) lies between points \(A\) and \(B\), the whole line segment \(AB\) is equal to the sum of segments \(AC\) and \(CB\).
\[AB = AC + CB\]
Use the Given Condition
It is given that
\[AC = CB\]
Substitute \(CB\) by \(AC\) in the equation \[ AB = AC + CB \]
Therefore,
\[AB = AC + AC\]
Simplify the Equation
Adding like terms
\[AB = 2AC\]
Divide Both Sides by 2
\[\frac{AB}{2} = \frac{2AC}{2}\]
\[\frac{AB}{2} = AC\]
Rewriting,
\[AC = \frac{1}{2}AB\]

📝 Geometrical Interpretation ›

Since \[ AC = CB \] point \(C\) divides the line segment \(AB\) into two equal parts.

Therefore, point \(C\) is called the midpoint of the line segment \(AB\).

Each smaller segment is equal to half of the complete segment.

\[ AC = CB = \frac{1}{2}AB \]

Final Result

\[ AC = \frac{1}{2}AB \]

🎯 Exam Significance Exam Significance ›

Questions based on line segment addition are commonly asked in school examinations.
This question develops proof-writing and logical reasoning skills.
Understanding midpoint concepts is important for coordinate geometry and constructions.
Competitive examinations often test the ability to apply simple geometric relations carefully.

← Q3

4 / 7 · 57%

Q5 →

📘 Concept & Theory Concept Used in this Question ›

This question is based on the concept of midpoint and uniqueness in geometry. A midpoint divides a line segment into two equal parts. The proof requires showing two things:

Existence of a midpoint.
Uniqueness of the midpoint.

Concept	Explanation
Line Segment	A part of a line bounded by two end points.
Midpoint	A point dividing a line segment into two equal parts.
Equal Segments	Segments having the same length.
Uniqueness	Only one such object exists satisfying the given condition.
Proof by Contradiction	Assuming the opposite statement and obtaining an impossibility.

Solution Strategy

Draw the line segment \(AB\).
Consider a point \(C\) on \(AB\) such that \[ AC = CB \]
Show that such a point acts as midpoint.
Assume another midpoint \(D\) exists.
Use logical reasoning to prove that \(C\) and \(D\) must coincide.

📊 Graph / Figure Graph / Figure ›

Point \(C\) divides the line segment \(AB\) into two equal parts.

✏️ Solution Complete Solution ›

Step-by-step Solution · 40 steps

Given
A line segment \[ AB \] is given.
To Prove
There exists one and only one point \(C\) on \(AB\) such that\[AC = CB\]
Proof
Existence of Midpoint
Consider the line segment\[AB\]
Let there exist a point \(C\) on the segment \(AB\) such that
\[AC = CB\]
Since point \(C\) divides the segment into two equal parts, point \(C\) is called the midpoint of the segment \(AB\).
Hence, a midpoint exists.
Assume Another Midpoint Exists
Suppose there exists another point \(D\) on the same line segment \(AB\) such that
\[AD = DB\]
Then point \(D\) is also a midpoint of \(AB\).
Now we shall prove that \(C\) and \(D\) cannot be different points.
Consider Relative Positions of \(C\) and \(D\)
Since both points lie on the same line segment \(AB\), either:
- \(C\) and \(D\) coincide, or
- one point lies between the other point and an endpoint.
Assume that \(C\) and \(D\) are distinct points.
Without loss of generality, let \(D\) lie between \(C\) and \(B\).
Compare Segment Lengths
Since \(D\) lies between \(C\) and \(B\), we have:
\[AD > AC\]
Also,
\[DB < CB\]
But from the midpoint conditions:
\[AC = CB\]
and
\[AD = DB\]
This is impossible because one equality implies a larger segment equals a smaller segment.
Hence, our assumption that \(C\) and \(D\) are different points is false.
Conclude Uniqueness
Therefore, \[C=D\]
Thus, there cannot be two different midpoints of the same line segment.
Final Conclusion
Every line segment has:
- one midpoint, and
- only one midpoint.
Therefore, every line segment has one and only one midpoint.

🎯 Exam Significance Exam Significance ›

Midpoint properties are frequently used in geometry proofs.
This question improves logical reasoning and proof-writing skills.
The concept of uniqueness appears in coordinate geometry and constructions.
Competitive examinations often include proof-based geometry questions.

← Q4

5 / 7 · 71%

Q6 →

📘 Concept & Theory Concept Used in this Question ›

This question is based on addition and subtraction of line segments. We use Euclid’s Common Notion:

“If equals are subtracted from equals, the remainders are equal.”

Concept	Explanation
Collinear Points	Points lying on the same straight line.
Segment Addition	The whole segment equals the sum of its parts.
Equal Segments	Segments having the same length.
Euclid’s Common Notion 3	If equals are subtracted from equals, the remainders are equal.

Solution Strategy

Observe the order of points on the straight line.
Express larger segments in terms of smaller segments.
Substitute the given equality: \[ AC = BD \]
Subtract equal quantities from both sides.
Obtain the required result step-by-step.

📊 Graph / Figure Graph / Figure ›

Points \(A,\; B,\; C,\; D\) lie on the same straight line in this order.

Fig. 5.10

✏️ Solution Complete Solution ›

Step-by-step Solution · 24 steps

Given
Points \(A,\; B,\; C,\; D\) are collinear in the order:
\[A - B - C - D\]
Also,
\[AC = BD\]
To Prove
\[AB = CD\]
Proof
Express Segment \(AC\)
Since point \(B\) lies between points \(A\) and \(C\), the whole segment \(AC\) is equal to the sum of segments \(AB\) and \(BC\)
\[AC = AB + BC\]
Express Segment \(BD\)
Since point \(C\) lies between points \(B\) and \(D\), the whole segment \(BD\) is equal to the sum of segments \(BC\) and \(CD\)
\[BD = BC + CD\]
Use the Given Condition
\[AC = BD\]
Substitute the values of \(AC\) and \(BD\):
\[AB + BC = BC + CD\]
Subtract Equal Quantities \(BC\) from both sides
\[AB + BC - BC = BC + CD - BC\]
Simplifying both sides:
\[AB = CD\]
By Euclid’s Common Notion 3:
“If equals are subtracted from equals, the remainders are equal.”

💡 Answer Final Answer ›

Final Result: \(AB = CD\)

🎯 Exam Significance Exam Significance ›

Questions based on segment addition are frequently asked in board examinations.
This problem develops algebraic reasoning in geometry.
Euclid’s common notions are important for proof-writing questions.
Competitive examinations often test logical simplification of geometrical relations.

← Q5

6 / 7 · 86%

Q7 →

📘 Concept & Theory Concept Used in this Question ›

This question is based on Euclid’s axioms and the meaning of universal truth. A universal truth is a statement that remains valid in every situation, not only in mathematics but also in everyday life.

Term	Meaning
Axiom	A statement accepted as true without proof.
Universal Truth	A fact that is true everywhere and in every context.
Whole	The complete object or quantity.
Part	A smaller portion of the whole.

🗺️ Solution Roadmap Step-by-step Plan ›

Recall Euclid’s Fifth Axiom.
Understand the meaning of “whole” and “part”.
Observe examples from geometry and daily life.
Explain why the statement is true universally.

📊 Graph / Figure Graph / Figure ›

Whole Line Segment

Part of the Segment

The complete segment \(AB\) is greater than its smaller part \(AC\).

✏️ Solution Complete Solution ›

Step-by-step Solution · 9 steps

Meaning of the Axiom
Euclid’s Fifth Axiom states:
“The whole is greater than the part.”
This means that if an object is divided into smaller pieces, then each individual piece is smaller than the complete object.
Understanding Through Geometry
Consider a line segment \[AB\]
If point \(C\) lies between \(A\) and \(B\), then segment \[ AC \] is only a part of the complete segment \[ AB \]
Therefore,
\[AB > AC\]

🎯 Exam Significance Exam Significance ›

Questions based on Euclid’s axioms are frequently asked in CBSE examinations.
Understanding universal truths strengthens logical reasoning.
Axioms form the foundation of mathematical proofs and deductive geometry.
Competitive examinations often test conceptual clarity rather than calculations.

← Q6

7 / 7 · 100%

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🎓

Chapter Complete!

All 7 solutions for INTRODUCTION TO EUCLID’S GEOMETRY covered.

↑ Review from the top

INTRODUCTION TO EUCLID’S GEOMETRY

There are an infinite number of lines which pass through two distinct points.

A terminated line can be produced indefinitely on both the sides.

If two circles are equal, then their radii are equal.

In Fig. 5.9, if AB = PQ and PQ = XY, then AB = XY.

Solution Strategy

parallel lines

Perpendicular Lines

Line Segment

Radius of a Circle

Square

How to Solve Questions Based on Postulates

Postulate (i)

Postulate (ii)

Final Conclusion

Solution Strategy

Final Result

Given

To Prove

Proof

Final Conclusion

Solution Strategy

Given

To Prove

Proof

Whole Line Segment

Part of the Segment

Chapter Complete!

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