Question 1

Question 1Which 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) In figure, if AB = PQ and PQ = XY, then AB = XY.

Accepted Answer

(i) FalseReason — There are infinite no. of lines that passes through a single point.(ii) FalseReason — By Euclid's axiom : Given two distinct points, there is a unique line that passes through them.(iii) TrueReason — By Euclid's postulate : A terminated line can be produced indefinitely on both side.(iv) TrueReason — If two circle are equal then there radii are equal, because if two circle are equal then on superimposing them their center and boundaries coincides and inscribe equal area thus their radii are equal.(v) TrueReason — According to Euclid's First Axiom,"Things which are equal to the same thing are equal to one another".Since, AB = PQ and PQ = XY∴ AB = XY.

Question 2

Question 2Give 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

Accepted Answer

We have to define some other terms like : Ray, Straight line and a Point.Ray - A ray is a part of a line, that has a fixed starting point but no end point. It can extend infinitely in one direction.Straight Line - A line that connects two points in a plane and extends to infinity in both directions. It is considered one dimensional figure.Point - A small dot made by a sharp pencil on a sheet of paper gives an idea about a point. A point has no dimension i.e, length, breadth and height, it has only a position.(i) If the perpendicular distance between two lines is constant, then these lines are considered parallel to each other.(ii) If the angle between two lines is equal to 90°, then these lines are considered perpendicular to each other.(iii) A part of a line that has two endpoints and is the shortest distance between them, is a line segment.(iv) The distance from the center to any point on the circumference of the circle is called the radius of the circle.(v) A square is a regular quadrilateral that has four equal sides and angle between each side is a right angle.

Question 3

Question 3Consider 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.

Accepted Answer

Using Euclid’s axioms to check these postulates :Yes, these postulates contain undefined terms like point and line.These two statements are consistent as they talk about two different situations meaning different things.These statements do not follow Euclid’s postulates but one of the axioms about “Given any two points, a unique line that passes through them” is followed.

Question 4

Question 4If a point C lies between two points A and B such that AC = BC, then prove that AC = 12\dfrac{1}{2}21​ AB. Explain by drawing the figure.

Accepted Answer

According to Euclid's axioms, we know that when equals are added to equals, the wholes are equal.Given : AC = BCAdding AC on both sides, we get⇒ AC + AC = BC + AC (BC + AC coincides with AB)⇒ 2 AC = AB⇒ AC = 12\dfrac{1}{2}21​ AB⇒ Hence, proved that AC = 12\dfrac{1}{2}21​ AB.

Question 5

Question 5In 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.

Accepted Answer

Let us consider that line segment AB has two midpoints C and D.Let's assume C to be the mid-point of AB.∴ AC = BCAdding AC on both sides, we get :⇒ AC + AC = BC + AC⇒ 2AC = AB⇒ AC = 12AB\dfrac{1}{2}AB21​AB .............(1)Let's assume that D is another mid-point of AB.∴ AD = BDAdding equal length AD on both sides, we get⇒ AD + AD = BD + AD (BD + AD coincides to AB)⇒ 2AD = AB⇒ AD = 12AB\dfrac{1}{2}AB21​AB .........(2)From equations (1) and (2), we get :⇒ AC = AD.⇒ C has to coincide with D for AC to be equal to AD.According to Euclid's Axiom 4: Things which coincide with one another are equal to one another.Hence, proved that a line segment has only one midpoint.

Question 6

Question 6In figure, if AC = BD, then prove that AB = CD.

Accepted Answer

Given, AC = BDFrom figure,⇒ AB + BC = BC + CD .........(1)Subtracting BC from both sides in equation (1)⇒ AB + BC - BC = BC + CD - BC⇒ AB = CDHence, proved that AB = CD.

Question 7

Question 7Why is Axiom 5, in the list of Euclid’s axioms, considered a ‘universal truth’? (Note that the question is not about the fifth postulate).

Accepted Answer

Axiom 5 states that :'Whole is always greater than its part'. This is a 'universal truth' because it holds true in any field of mathematics and in other disciplinarians of science as well.Let us take two cases: one in the field of mathematics and one other than that.Case 1 — Let t represent a whole quantity and only a, b, c are parts of it.Such that :t = a + b + cClearly, t will be greater than all of its parts a, b and c. Therefore, it is rightly said that the whole is greater than the part.Case 2 — Let us consider continent Asia. Then, let us consider a country India which belongs to Asia. India is a part of Asia and it can also be observed that Asia is greater than India. That is why we can say that the whole is greater than the part.

Introduction to Euclid's Geometry