RowQ
The Vault
RowQ
The Vault
CBSE Class 9 Maths · 11 questions · 26 marks
Long before formulas, geometry began as an argument: start from a few statements everyone accepts, then prove everything else. Euclid wrote those starting statements down as definitions, axioms and postulates, and this chapter shows you how the whole structure hangs together. It is less about calculating and more about learning what counts as a proof.
Which of the following is Euclid's third postulate?
Answer
A circle can be drawn with any centre and any radius. This is Euclid's third postulate, which guarantees that the compass construction is always available. The statement about right angles is the fourth postulate, producing a terminated line is the second postulate, and "the whole is greater than the part" is an axiom (common notion), not a postulate.
The difference between an axiom and a postulate in Euclid's system is that:
Answer
Axioms are assumptions used across all of mathematics while postulates are assumptions specific to geometry. Euclid called his general common notions, such as "the whole is greater than the part", axioms because they apply to magnitudes of every kind. The five statements about points, lines and circles were called postulates because they concern geometry alone. Neither axioms nor postulates are proved — both are accepted without proof, which rules out the first and third options.
If a point M lies between points L and N on a line segment such that LM = MN, then M is called:
Answer
M is the midpoint of LN. A point lying between L and N that divides the segment into two equal parts is by definition the midpoint. Euclid's axioms guarantee this point is unique: if there were two such points, subtracting equals from equals would force them to coincide.
Playfair's axiom states that through a point not lying on a given line:
Answer
Exactly one line parallel to the given line can be drawn. Playfair's axiom is an alternative, more convenient statement of Euclid's fifth postulate, and the two are logically equivalent. It asserts both existence (at least one parallel exists) and uniqueness (no more than one).
Assertion (A): Euclid's fifth postulate is equivalent to the statement that there exists a pair of straight lines that are everywhere equidistant from one another. Reason (R): Euclid's fifth postulate concerns interior angles formed when a straight line falls on two straight lines.
Answer
Both A and R are true but R is not the correct explanation of A. Assertion A is true: the existence of everywhere-equidistant lines is one of the several known statements logically equivalent to the fifth postulate. Reason R is also true, since the fifth postulate is stated in terms of interior angles summing to less than two right angles. However, R merely describes the wording of the postulate; it does not by itself explain why the equidistance statement is equivalent to it, so R is not the correct explanation of A.
State any two of Euclid's axioms (common notions) and give one everyday illustration of either.
Answer
Axiom 1: Things which are equal to the same thing are equal to one another. Axiom 2: If equals are added to equals, the wholes are equal. Illustration of Axiom 1: if Riya's ribbon is the same length as a metre scale, and Sam's ribbon is also the same length as that metre scale, then Riya's and Sam's ribbons must be equal in length to each other.
If A, B and C are three points on a line such that B lies between A and C, prove that AB + BC = AC.
Answer
Since B lies between A and C, the segment AC is made up of the two segments AB and BC placed end to end with no overlap and no gap. By Euclid's axiom that the whole is equal to the sum of its parts, the length of the whole segment AC equals the sum of the lengths of its parts. Therefore AB + BC = AC. This also shows AC > AB, by the axiom that the whole is greater than the part.
Why is it impossible to define every term in geometry? Which terms are therefore left undefined?
Answer
Every definition must use other, already-understood terms. If we insisted on defining everything, each definition would need earlier definitions, leading either to an endless chain or to circular reasoning where a term is eventually defined using itself. To avoid this, geometry begins with a few undefined terms — point, line and plane — whose properties are described by the axioms and postulates rather than by formal definitions.
(a) State Euclid's fifth postulate in your own words. (b) Explain how it differs in character from his first four postulates. (c) State Playfair's axiom and explain why mathematicians often prefer it.
Answer
(a) If a straight line crosses two other straight lines and the two interior angles formed on one side of it add up to less than 180°, then those two lines, when extended far enough, will meet on that side. (b) The first four postulates are short, immediately obvious statements about drawing lines, extending them, drawing circles and comparing right angles. The fifth is far longer, more complicated, and not self-evident at a glance; it also makes a claim about lines extended indefinitely, which cannot be checked by any drawing. For these reasons mathematicians spent centuries unsuccessfully trying to prove it from the other four, and their failure eventually led to the discovery of consistent non-Euclidean geometries. (c) Playfair's axiom states that through any point not on a given line, exactly one line can be drawn parallel to that line. It is preferred because it is much easier to state and to picture than the fifth postulate, while being logically equivalent to it — assuming either one lets you prove the other.
(a) Using Euclid's axioms, prove that if two line segments PQ and RS are each equal to a third segment TU, then PQ = RS. (b) Prove that a line segment has exactly one midpoint.
Answer
(a) We are given PQ = TU and RS = TU. By Euclid's first axiom, things which are equal to the same thing are equal to one another. Here both PQ and RS are equal to the same segment TU. Therefore PQ = RS. (b) Let AB be a line segment and suppose, for contradiction, that it has two midpoints M and N with M ≠ N. Since M is a midpoint, AM = (1/2)AB. Since N is a midpoint, AN = (1/2)AB. By the first axiom, AM = AN, because both are equal to the same quantity (1/2)AB. But M and N both lie on segment AB measured from the same endpoint A, so equal distances from A along the same segment force M and N to be the same point. This contradicts our assumption that M ≠ N. Hence a line segment has exactly one midpoint.
Read the following and answer the questions that follow: A class is asked to lay out a rectangular vegetable plot using only a rope and pegs, exactly as ancient surveyors would have. They stretch the rope between two pegs to make a straight boundary, swing the rope around a fixed peg to mark a curved edge, and extend one boundary further to reach the fence. (a) Which Euclidean postulate justifies stretching the rope between two pegs? (b) Which postulate justifies swinging the rope around a fixed peg? (c) Which postulate justifies extending the boundary further? (d) The teacher says the two shorter sides of the plot are each equal to the rope's length, and concludes they are equal to each other. Which axiom is being used?
Answer
(a) Euclid's first postulate: a straight line may be drawn from any one point to any other point. The two pegs are the two points, and the taut rope is the straight line joining them. (b) Euclid's third postulate: a circle can be drawn with any centre and any radius. The fixed peg is the centre and the rope's length is the radius. (c) Euclid's second postulate: a terminated line (line segment) can be produced indefinitely on either side. Extending the boundary to the fence is exactly this. (d) Euclid's first axiom: things which are equal to the same thing are equal to one another. Both shorter sides equal the rope's length, so they equal each other.
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