Exercise-5.1, Class 9th, Maths, Chapter 5, NCERT

Q1. Which statements are true / false? Give reasons.

(i) Only one line can pass through a single point.
False. A single point does not determine a unique line — infinitely many lines (with different directions) can pass through the same point.

(ii) There are an infinite number of lines which pass through two distinct points.
False. By Axiom 5.1 (or the usual Euclidean axiom) exactly one line passes through two distinct points.

(iii) A terminated line can be produced indefinitely on both the sides.
True. This is exactly Euclid’s Postulate 2: a line segment (terminated line) can be extended indefinitely to a line.

(iv) If two circles are equal, then their radii are equal.
True. Two circles are equal (congruent) exactly when their radii are equal; equal circles have equal radii.

(v) In Fig. 5.9, if AB = PQ and PQ = XY, then AB = XY.
True. This follows from Euclid’s common notion: “Things which are equal to the same thing are equal to one another.”

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Q2. Give definitions. Are there other terms that need definition first?

(Note: in Euclidean development some primitive/undefined terms are point, line, plane. These are taken as basic; other definitions use them.)

(i) Parallel lines
Two lines in the same plane that do not meet (do not intersect) no matter how far they are extended.

(ii) Perpendicular lines
Two lines that meet to form a right angle (an angle of ${90}^{\circ }$).

(iii) Line segment
Part of a line bounded by two distinct endpoints; the set of points between and including those endpoints.

(iv) Radius of a circle
A line segment joining the centre of the circle to any point on the circle.

(v) Square
A quadrilateral with four equal sides and four right angles.

Other terms needed first: point, line, plane, angle, right angle — many of these are treated as primitive/undefined or assumed known before giving the above definitions.

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Q3. Consider these two postulates:

(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.

For each: do they contain undefined terms? are they consistent? do they follow from Euclid’s postulates?

Answer & explanation

• Both statements use the undefined term point and the notion between / on the same line (the concept “between” is not one of Euclid’s five postulates and is effectively an order relation that is not defined from those five). So yes, they involve undefined/primitive notions (point, line, between).

• Consistency: Both are consistent (they state plausible geometric facts) — they do not contradict Euclid’s postulates. In fact they are typical order/axiom additions used in modern axiom systems (they assert betweenness and existence of non-collinear points).

• Do they follow from Euclid’s five postulates? No. Euclid’s original five postulates do not guarantee a point between two given points nor do they assert the existence of three non-collinear points. Those facts are independent of the five postulates and are usually added as additional axioms (order and incidence axioms) in modern axiom systems. In short: they do not follow from Euclid’s five postulates alone.

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Q4. If a point $C$ lies between two points $A$ and $B$ such that $AC=BC$, then prove that $AC=\frac{1}{2}AB$

Proof. Since  is between $A$ and $B$, by definition $AB=AC+CB$. Given $AC=BC$, so $AB=AC+AC=2\cdot AC$.

Hence $AC={\displaystyle \frac{1}{2}}AB$

(Also illustrated by a simple line diagram with A—C—B.)

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Q5. In Question 4, point $C$ is called a midpoint of line segment $AB$. Prove that every line segment has one and only one midpoint.

Existence. Let segment $AB$ be given. Construct two circles: one with centre $A$ and radius $AB$, and another with centre $B$ and radius $BA$. These two circles meet at two points (by Postulate 3 and the standard construction). Join those intersection points — their joining line is the perpendicular bisector of $AB$. Where that perpendicular bisector meets $AB$ is a point $M$. By construction $AM=MB$. Thus a midpoint exists.

Uniqueness. Suppose a segment $AB$ had two distinct midpoints $C$ and $D$. Then $AC=CB$ and $AD=DB$. From these, $AB=AC+CB=2AC$ and also $AB=AD+DB=2AD$. Hence $2AC=2AD$, so $AC=AD$. But on a line through $A$ the distance from $A$ to different points increases strictly with the point’s position along the ray; two distinct points on segment $AB$ cannot be at the same distance from $A$. Therefore $C$ and $D$ cannot be distinct; they must coincide. So the midpoint is unique. □

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Q6. In Fig. 5.10, if $AC=BD$, then prove that $AB=CD$

(Using the order shown in the figure: the four points are collinear in the order $A$–$B$–$C$–$D$.)

Because the points are in order $A\text{\hspace{0.17em}⁣}-\text{\hspace{0.17em}⁣}B\text{\hspace{0.17em}⁣}-\text{\hspace{0.17em}⁣}C\text{\hspace{0.17em}⁣}-\text{\hspace{0.17em}⁣}D$:

• $AC=AB+BC$.

• $BD=BC+CD$.

Given $AC=BD$, equate the two expressions:

$$AB+BC=BC+CD\mathrm{.}$$

Cancel $BC$ from both sides to get $AB=CD$.

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Q7. Why is Axiom 5 in Euclid’s list considered a “universal truth”? (Note: not the fifth postulate.)

Axiom 5 (one of Euclid’s common notions) states: “Things which coincide with one another are equal to one another.”
This is considered a universal truth because it expresses a self-evident identity principle: if two things exactly coincide (i.e., occupy the same position or are identical in every respect), they must be equal. It is not specific to geometry — it applies to magnitudes and objects in general mathematics (hence “common notion”). It formalizes the intuitive idea of equality by superposition: identical objects are equal. That self-evidence makes it a universal/primitive assumption used to build further proofs.