CBSE Class 9 Mathematics Introduction to Euclids Geometry Assignment Set 02

Question. If Cis t he midpoint of the segment AB, P and Q are midpoints of the segments AC and BC respectively, find BQ.
(A) 1/2 AB
(B) 1/3 AB
(C) 1/4 AB
(D) 1/5 AB
Answer : C

Question. If Cis the midpoint of the line segment AB, D and E are midpoints of the segments AC and BC respectively, what is the length of AB?
(A) 2 AD
(B) 4 BC
(C) 4 BE
(D) 2 CD
Answer : C

Question. Which of the following describe a surface?
(A) Lengt h and Breadt h
(B) Lengt h only
(C) Breadth only
(D) Length and Height
Answer : A

Question. How many point s does a line cont ain?
(A) Two
(B) Three
(C) Four
(D) Infinitely many
Answer : D

Question. If AC = PQ and CP = BQ, find the midpoint of the line segment AB.
(A) p
(B) Q
(C) C
(D) B
Answer : A

Question. In the figure given, if AC = BD, what is the measure of AB?

(A) AD
(B) BD
(C) CD
(D) AC
Answer : C

Question. If /, m and n are three distinct lines such that l║m and l║n , which of the following holds good?
(A) m ⊥ n
(B) m║n
(C) m = n
(D) l ⊥ n
Answer : B

Question. If a point C lies in between A and B, what is AC + BC equal to?
(A) AB
(B) 1/2 AB
(C) 2 AB
(D) 4 AB
Answer : A

Question. If l║m and l ⊥ n, identify the correct option.
(A) l║n
(B) m ⊥ n
(C) m║n
(D) m ⊥ l
Answer : B

Question. In the figure given, if A and B are the centres of the two intersecting circles, what type of a triangle is Δ ABC ?

(A) A scalene triangle
(B) A right triangle
(C) An isosceles t riangle
(D) An equilat eral triangle
Answer : D

Question. What are the boundaries of surfaces?
(A) Surfaces
(B) Angles
(C) Lines
(D) Points
Answer : C

Question. In ancient India, what are the shapes of alt ars used for house hold rituals?
(A) Squares and circles
(B) Triangles and rect angles
(C) Trapeziums and pyramids
(D) Rectangles and squares
Answer : A

Question. Which of the following is stated in the form 'Lines are parallel if they do not intersect '?
(A) An axiom
(B) A definition
(C) A postulate
(D) A proof
Answer : B

Question. In Indus Valley Civilisation (about300 B.C), what were the dimensions of the bricks used for construction work?
(A) 1 :3:4
(B) 4:2:1
(C) 4:4: 1
(D) 4:3:2
Answer : B

Introduction to Euclid's Geometry

Q 1 Write Ecluid's definition of straight line.
Answer: According to Euclid, a straight line is a line which lies evenly with the points on itself.
In simple words: A straight line is perfectly even and flat without any curves.

Exam Tip: Memorize exact definitions from Euclid's Elements as they are frequently asked in 1-mark questions.

Q 2 State true of false : Two distinct lines intersect at more than one point.
Answer: False
In simple words: Two different lines can only cross each other at one single point.

Exam Tip: Recall Euclid's axiom that two distinct intersecting lines cannot have more than one point in common.

Q 3 Fill in the blank : A_____ is that which has no part.
Answer: point
In simple words: A point is the fundamental building block of geometry that has no length, breadth, or thickness.

Exam Tip: Keep track of Euclid's initial definitions, especially for point, line, and surface.

Q 4 How many points a line segment can have?
Answer: Infinitely many points.
In simple words: A line segment is made up of an infinite number of closely packed points.

Exam Tip: Remember that any line or line segment contains an infinite set of points.

Q 5 State true or false: Given two distinct points, there are two lines which passes through them.
Answer: False
In simple words: Only one unique straight line can pass through any two given distinct points.

Exam Tip: This corresponds to Euclid's first postulate.

Q 6 Fill in the blank : Three or more lines are said to be _____if their common point lies on them.
Answer: concurrent
In simple words: Lines that pass through the exact same single point are called concurrent lines.

Exam Tip: Do not confuse concurrent lines (passing through one point) with collinear points (lying on one line).

Q 7 According to Euclid, Name of geometrical figure which has only length and breadth.
Answer: Surface (or plane surface)
In simple words: A surface has length and breadth, but no thickness.

Exam Tip: Solid has length, breadth, and thickness; surface has length and breadth; line has only length; point has none.

Q 8 State two equivalent versions of Ecluid's fifth postulate.
Answer: 1. Playfair's Axiom: For every line \(l\) and for every point \(P\) not lying on \(l\), there exists a unique line \(m\) passing through \(P\) and parallel to \(l\).
2. Two distinct intersecting lines cannot be parallel to the same line.
In simple words: Alternative ways of stating parallel line properties that are equivalent to Euclid's parallel postulate.

Exam Tip: Playfair's Axiom is the most commonly used equivalent version in exams.

Q 9 If A, B and C are three points on a line, and B lies between A and C then prove that AB + BC = AC.
Answer: Given that point B lies between A and C.
From the figure of line segments, the portion AB and the portion BC together coincide with the line segment AC.
Using Euclid's axiom "Things which coincide with one another are equal to one another", we get:
\(AB + BC = AC\)
Hence proved.
In simple words: Adding two adjacent parts of a line segment gives the total length of the segment.

Exam Tip: Mention the relevant Euclid axiom when proving geometric relations.

Q 10 If a point C lies between two points A and B such that AC = BC, then prove that AC = (1/2)AB.
Answer: Given that point C lies between A and B, so:
\(AC + BC = AB\)
Given that \(AC = BC\), substitute BC with AC:
\(AC + AC = AB\)
\(2AC = AB\)
Dividing both sides by 2:
\(AC = \frac{1}{2}AB\)
Hence proved.
In simple words: If a point divides a line segment into two equal halves, each half is half the total length.

Q 11 Which axiom is related to comparison of things According to Euclid’s axiom?
Answer: Euclid's Axiom: "Things which coincide with one another are equal to one another" and "The whole is greater than the part" are used for comparison.
In simple words: Axioms that compare quantities by fitting them together or checking sizes.

Exam Tip: Quote the exact axiom statements for full credit.

Q 12 If a point C lies between two points A and B such that AC = BC, then find the relation between BC and AB.
Answer: Given \(AC = BC\) and \(AC + BC = AB\).
Since \(AC = BC\), we can write:
\(BC + BC = AB\)
\(2BC = AB\)
\(BC = \frac{1}{2}AB\)
In simple words: BC is also half of the total length AB.

Exam Tip: Follow the same substitution method as Q 10.

Q 13 As per Euclid’s axiom, ‘If equals are added to equals, explain with examples.
Answer: Euclid's Axiom states: "If equals are added to equals, the wholes are equal."
Example: If \(x = y\), then adding 5 to both sides gives \(x + 5 = y + 5\). Similarly, if two persons have equal weight and both gain 2 kg, their new weights are equal.
In simple words: Starting with equal amounts and adding the same quantity to both keeps them equal.

Exam Tip: Always provide a mathematical or real-life example when asked to explain an axiom.

Q 14 What are the types of the boundaries of the surfaces?
Answer: The boundaries of surfaces are curves or straight lines.
In simple words: Edges of a surface are formed by lines or curves.

Q 15 Which proof was given by the great mathematician Thales about circle?
Answer: Thales proved that a circle is bisected (divided into two equal parts) by its diameter.
In simple words: Any diameter cuts a circle into two identical halves.

Exam Tip: Greek contributions like Thales' theorems are important historical context in this chapter.

Q 16 Which type of the the shape of altars used for household rituals in the Vedic period?
Answer: Altars used for household rituals (grihya rituals) were shaped as squares and circles.
In simple words: Vedic altars for homes were built using squares and circles.

Q 17 In Fig., if AC = BD then prove that AB = CD.
Answer: Given \(AC = BD\).
From the figure, segment \(AC = AB + BC\).
Segment \(BD = BC + CD\).
Substituting these into the given equation:
\(AB + BC = BC + CD\)
Subtracting BC from both sides (using Euclid's axiom: "If equals are subtracted from equals, the remainders are equal"):
\(AB = CD\)
Hence proved.
In simple words: If two overlapping segments are equal, removing their common overlapping part leaves the remaining parts equal.

Exam Tip: This is a standard 3-mark textbook proof; write every axiom used.

Q 18 If PS = RT as shown in the figure, then what will be the value of ST?
Answer: Given \(PS = RT\) and length of segment \(RS = 3\) cm, total \(PT = 5\) cm.
From figure, \(PS = PR + RS\)
\(RT = RS + ST\)
Since \(PS = RT\), we have \(PR + RS = RS + ST\), which implies \(PR = ST\).
Also, given \(PT = 5\) cm and \(PS = RT\)...
Using total lengths: \(ST = PT - PS\) ... wait, let us use segment addition: \(PT = PR + RS + ST = 5\).
Given \(PS = RT \implies PR + RS = RS + ST \implies PR = ST\).
Also \(PS = PR + 3\), \(RT = 3 + ST\).
Since \(PS = RT\), \(PR = ST\).
Total length \(PT = PR + RS + ST = 5\).
Since \(PR = ST\), \(ST + 3 + ST = 5 \implies 2ST + 3 = 5 \implies 2ST = 2 \implies ST = 1\) cm.
In simple words: By equating overlapping segments, we find the length of ST is 1 cm.

Exam Tip: Break down total lengths into individual sub-segments before solving numerical geometry problems.

Q 19 Define the following according to book ’Element’ by Euclid: (i) Surface (ii) Point (iii) Straight line (iv) Line.
Answer: (i) Surface: A surface is that which has length and breadth only.
(ii) Point: A point is that which has no part.
(iii) Straight line: A straight line is a line which lies evenly with the points on itself.
(iv) Line: A line is breadthless length.
In simple words: Euclid's exact definitions for the basic geometric terms.

Exam Tip: Memorize all definitions word-for-word as given in NCERT.

Q 20 What is Playfair’s Axiom?
Answer: Playfair's Axiom states that "For every line \(l\) and for every point \(P\) not lying on \(l\), there exists a unique line \(m\) passing through \(P\) and parallel to \(l\)."
In simple words: Through a point outside a line, exactly one parallel line can be drawn.

Exam Tip: This is a very important alternative to Euclid's fifth postulate.

Q 21 What are two equivalent versions of Euclid’s fifth postulate?
Answer: 1. Playfair's Axiom.
2. Two distinct intersecting lines cannot be parallel to the same line.
In simple words: Two standard statements equivalent to the parallel postulate.

Exam Tip: Be prepared to state both versions clearly.

Q 22 Define Postulate according to Euclid.
Answer: A postulate is a geometric assumption that is specific to geometry and accepted as true without proof.
In simple words: Basic geometric rules assumed to be true without needing proof.

Exam Tip: Distinguish between axioms (general assumptions across all mathematics) and postulates (geometry-specific assumptions).

Q 23 State Euclid’s five postulates.
Answer: Postulate 1: A straight line may be drawn from any one point to any other point.
Postulate 2: A terminated line can be produced indefinitely.
Postulate 3: A circle can be described with any center and any radius.
Postulate 4: All right angles are equal to one another.
Postulate 5: 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.
In simple words: Euclid's five fundamental rules for constructing geometric figures.

Q 24 Define the following terms: (i) Intersecting lines (ii) Parallel lines (iii) Line segment (iv) Collinear points.
Answer: (i) Intersecting lines: Two lines that share a common point are called intersecting lines.
(ii) Parallel lines: Two lines in a plane that do not intersect anywhere, maintaining a constant perpendicular distance, are parallel lines.
(iii) Line segment: A part of a line with two definite end points is called a line segment.
(iv) Collinear points: Three or more points that lie on the same straight line are called collinear points.
In simple words: Basic geometric definitions for lines and points.

Exam Tip: Always include key conditions like "two end points" for line segments and "same line" for collinear points.

Q 25 Define the following terms: (i) Axiom (ii) Theorem
Answer: (i) Axiom: Assumptions that are obvious universal truths accepted across all branches of mathematics without proof.
(ii) Theorem: Statements that are proved using definitions, axioms, previously proved statements, and logical reasoning.
In simple words: Axioms are accepted without proof, while theorems must be proved.

Q 26 Give seven Euclid’s axioms.
Answer: 1. Things which are equal to the same thing are equal to one another.
2. If equals are added to equals, the wholes are equal.
3. If equals are subtracted from equals, the remainders are equal.
4. Things which coincide with one another are equal to one another.
5. The whole is greater than the part.
6. Things which are double of the same things are equal to one another.
7. Things which are halves of the same things are equal to one another.
In simple words: Euclid's seven general rules of mathematical logic.

Exam Tip: Memorize at least 5-7 axioms verbatim.

Q 27 Mention Five postulates of Eulid.
Answer: Refer to Q 23 for the complete statement of Euclid's five postulates.
In simple words: The five basic construction rules of geometry.

Exam Tip: Same as Q 23.

Q 28 Prove that an equilateral triangle can be constructed on any given line segment.
Answer: Step 1: Let AB be the given line segment.
Step 2: With center A and radius AB, draw a circle using Postulate 3.
Step 3: With center B and radius BA, draw another circle.
Step 4: Let the two circles intersect at point C.
Step 5: Join AC and BC using Postulate 1.
Step 6: Since radius of first circle is AB, \(AC = AB\). Since radius of second circle is BA, \(BC = AB\).
Therefore, \(AC = BC = AB\), proving that triangle ABC is an equilateral triangle.
In simple words: Using circles with equal radii from both ends of a line segment forms an equilateral triangle.

Exam Tip: This is Proposition 1 from Euclid's Elements; write step-by-step construction using postulates.

Q 29 Give the definition for each of the following terms:- (i) Parallel lines, (ii) Perpendicular lines, (iii) Line segment, (iv) Radius of a circle, (v) Square.
Answer: (i) Parallel lines: Lines that never intersect and remain at a constant distance apart.
(ii) Perpendicular lines: Two intersecting lines that meet at a right angle (\(90^\circ\)).
(iii) Line segment: A portion of a line with two endpoints.
(iv) Radius of a circle: A line segment joining the center of a circle to any point on its circumference.
(v) Square: A quadrilateral with four equal sides and four right angles.
In simple words: Standard geometric definitions for shapes and lines.

Exam Tip: Include key geometric properties like right angles and equal sides in definitions.

Most Important Questions

Q 1 Define the following terms a) Point b) Line c) Line Segment
Answer: a) Point: That which has no part.
b) Line: Breadthless length.
c) Line Segment: A part of a line with two endpoints.
In simple words: Fundamental definitions of geometry.

Exam Tip: Essential 1-mark definitions.

Q 2 Explain the following terms : a) Concurrent lines b) Collinear points c) Parallel lines d) Intersecting lines
Answer: a) Concurrent lines: Three or more lines passing through the same point.
b) Collinear points: Three or more points lying on the same straight line.
c) Parallel lines: Lines in the same plane that never meet.
d) Intersecting lines: Lines that cross each other at a single common point.
In simple words: Relationships between multiple lines and points.

Exam Tip: Use diagrams mentally or on paper to verify definitions.

Q 3 How many lines can pass through a given point?
Answer: Infinitely many lines.
In simple words: An endless number of lines can pass through a single point.

Exam Tip: Contrast this with two points (only 1 line).

Q 4 How many lines can pass through two given points?
Answer: Exactly one unique line.
In simple words: Only one straight line can connect two points.

Exam Tip: This is Euclid's first postulate.

Q 5 How many line segments can pass through three collinear point A, B, C?
Answer: Three line segments (AB, BC, and AC).
In simple words: The segments formed by taking pairs of points among A, B, and C are AB, BC, and AC.

Exam Tip: Count carefully based on order of points.

Q 6 State True or false a) Two lines intersect at a point. b) A line segment has a fixed length. c) A ray has a fixed length. d) Only one line can pass through a given point. e) Two lines are coincident if they have only one point in common.
Answer: a) False (Two lines *may* intersect at a point, but not necessarily all lines).
b) True.
c) False (A ray has one endpoint and extends infinitely in the other direction, so it has no fixed length).
d) False (Infinitely many lines pass through one point).
e) False (Coincident lines have infinitely many points in common).
In simple words: True or false check for basic geometric properties.

Exam Tip: Read every statement carefully for absolute words like "only" or "always".

Q 7 Fill in the blanks: a)Two distinct points in a plane determine a. …line. b) Given a line and a point, which is not on the line, there is …… line, which passes through the given point and is …… to the given line. c) Whole of anything is …… to the sum of its parts and …… than any one of them. d) If equals are subtracted from wholes the remainders are …… .
Answer: a) unique (or one)
b) unique, parallel
c) equal, greater
d) equal
In simple words: Standard fill-in-the-blanks based on axioms and postulates.

Exam Tip: Learn axioms word-for-word to fill blanks accurately.

Q 8 State the two equivalent version of Euclid’s fifth postulate.
Answer: Refer to Q 8 of the main section (Playfair's Axiom and non-intersecting parallel lines property).
In simple words: Alternative statements for parallel lines.

Q 9 If C is a point which lies between two points A and B such that AC = BC, then prove that AC= ½ AB. Explain by drawing the line.
Answer: Refer to Q 10 solution for the algebraic proof. 
Explanation by drawing: Draw a straight line segment AB, and mark point C exactly in the middle such that length AC equals length BC.
In simple words: Midpoint divides a line segment into two equal halves.

Exam Tip: Always accompany line segment proofs with a neat geometric sketch.

Q 10 Prove that the mid-point of any line segment is unique.
Answer: Let AB be a line segment. Suppose it has two midpoints, C and D.
Since C is a midpoint of AB, \(AC = \frac{1}{2}AB\).
Since D is also a midpoint of AB, \(AD = \frac{1}{2}AB\).
Therefore, \(AC = AD\).
This is possible only if points C and D coincide with each other.
Thus, every line segment has one and only one (unique) midpoint.
In simple words: A line segment cannot have two different midpoints.

Exam Tip: Use proof by contradiction or direct equality with axioms to prove uniqueness.