Thursday, July 12, 2012

CBSE Class 9- CH5: Introduction to Euclid's Geometry

Fragment of Euclid's Elements
source: wikicommons
NCERT Chapter Solutions
and Other Q & A

Exercise 5.1

Q1: 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.
(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) As shown in Fig. below, if AB = PQ and PQ = XY, then AB = XY.
Answer:

(i) False. An infinite number of lines can pass through a single point to different directions.

(ii) False. Through two distinct points only one line can pass.

(iii) True. A line it can be extended indefinitely on both the sides.
(iv) True. If two circles are equal, their centres will coincide and inscribe equal areas(πr2). Thus their radii will also be equal.

(v) True. According to Euclid's first Axiom, "Things which are equal to the same thing are equal to one another."

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?
(i) parallel lines 
(ii) perpendicular lines 
(iii) line segment
(iv) radius of a circle 
(v) square

Answer:
(i) parallel lines: Two lines are said to be parallel lines, if the perpendicular distance between them is same. To define parallel lines, we must define, 'point', 'straight lines', 'point of intersection; and 'perpendicular distance'.


(ii) perpendicular lines : If two lines intersect at angles 90°, the lines are said to be perpendicular line. To define perpendicular lines, we must define, 'line', 'angle' and 'point of intersection'.


(iii) line segment: A straight line drawn from any point to any other point is called a line segment. To
define a line segment, we must define 'point' and 'line segment' first.

(iv) radius of a circle: It is the distance between the centre point of a circle to any point lying on the circle (circumference). We must predefine 'point', 'centre' and 'circle'.


(v) square: A square is a quadrilateral which has four equal sides in length and its adjacent sides intersect at right angles (90°). We must define, quadrilateral, side, point of intersection and angle.


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.

Answer:
(a) The postulate (i) contains undefined term i.e. 'between A and B'. It is unclear where point C lies, does it lie on line segment AB, or above AB or below AB or very far above AB. There are other undefined terms such as point, line etc.

(b) Both the postulates are consistent since they do not conflict each other and refer to two different situations.

(c) These postulates do not follow Euclid's postulates. However they follow his axioms. E.g. Axiom 5.1: Given two distinct points, there is a unique line that passes through them.

Q4: If a point C lies between two points A and B such that AC = BC, then prove that AC = AB/2. Explain by drawing the figure.

Answer:  AC = CB (Given)
Also AC + AC = BC + AC.              (Equals are added to equals)
∵ things which coincide with one another are equal to one another.
 
∴ AC + BC coincides with AB
⇒ 2AC = AB
⇒ AC = ½AB.

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.

Answer: Let there be two such mid points C and D. Then using above said  theorem (see answer 4), we can prove
AC = ½AB                                          ... (I)
and AD = ½AB                                    ... (II)
From I and II, we have
∴ AC = AD = ½AB
AC and AD can be equal only if D coincides with C. Therefore, C is the unique
mid-point.

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

Answer: Given, AC = BD
Also AC = AB + BC             (Point B lies on AC)
and BD = BC + CD             (Point C lies on BD)

∴ AB + BC = BC + CD
Using Euclid’s axiom, when equals are subtracted from equals, the remainders
are also equal.
⇒  AB + BC - BC = BC + CD - BC
⇒  AB - 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.)

Answer: Euclid's postulate 5 states, "The whole is greater than the part." It is considered 'universal truth', because it holds true in every field.

Consider the following cases:


Case I: Consider a group of numbers 15, 8, 4, 2, 1 such that 15 = 8 + 4 + 2 + 1 and 15 is greater than any of its part (8, 4, 2, 1)

Case II: Consider a circle, consisting of six sectors (a, b, c, d, e and f).
The area of a circle as a whole is greater than that of any sector (its part).

Q8: Name the mathematician who is credited for giving first known proof.

Answer: Thales

Q9: What is the first proof provided by Thales?

Answer: A circle is bisected by its diameter.

Q10: Name the treatise compiled by Euclid.

Answer: Elements. Originally written in Greek.
(Consists of 13 chapters. Written around 300 B.C. It is a systematic account of the geometry and number theory of that time.Later translated in Arabic, Latin, English and other languages. Considered to be the second most widely printed book after Bible.)

Q11: Who is known as Father of Geometry.

Answer: Euclid.

Watch the video on Euclid - Father of Geometry by KhanAcademy



Q12: Name the manuals on geometrical construction compiled during ancient India.

Answer: Sulbasutras (800 BC to 500 BC)

EXERCISE 5.2

Q13: How would you rewrite Euclid’s fifth postulate so that it would be easier to understand?

Answer:
(i) 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.


If we draw perpendicular both from L and M i.e. AB and XY. The perpendicular distances are equal i.e. AB = XY.

(ii) Two distinct intersecting lines cannot be parallel to the same line.

Q & A from Exam. papers and other books

Q14: Which geometric terms are considered 'undefined'?

Answer: Point, line, plane are taken as undefined terms.

Q15: The three steps from solid to point are:
(a) Solid - Surface - Line - Point
(b) Line - Point - Surface - Solid
(c) Surface - Point - Line - Solid
(d) Point - Surface - Line - Solid

Answer:  (a) Solid - Surface - Line - Point

Q16: Which of the following is an example of a geometrical line.
(a) Black board
(b) Sheet of paper
(c) Meeting place of two walls
(d) Tip of the sharp pencil.
Answer:  (c) Meeting place of two walls

Q17(CBSE 2011): Lines are parallel to each other if they do not intersect' is stated in the form of:
(a) a definition
(b) an axiom
(c) a postulate
(d) a proof

Answer: (c) a postulate


Q18(CBSE 2011): 'Two intersecting lines cannot be parallel to the same line' is stated in the form of:
(a) an axiom
(b) a definition
(c) a postulate
(d) a proof

Answer: (c) a postulate

Q19:  Which of the following needs a proof? (a) Axiom
(b) Postulate
(c) Definition
(d) Theorem

Answer: (a) Axiom (Interpretation of Playfair’s Axiom)


Q20: Plafair's Axiom is equivalent to which postulate of Euclid's geometry?

Answer: Euclid’s Fifth Postulate

Q21: What is great achievement of  Euclid’s Fifth Postulate

Answer: It lead to development of non-euclidean geometry.

Q22: Give any example of euclidean geometry.

Answer: Spherical geometry, longitudes latitudes etc.

Q23:Name mathematicians of ancient India who contributed to geometry.

Answer: Aryabhatta, Brahmgupta and Bhaskaracharya
 
Q24: The number of dimensions a surface has is ..... .(a) 0
(b) 1
(c) 2
(d) 3

Answer: (c) 2

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