CBSE Class 9 Maths Chapter 5: Introduction to Euclid's Geometry NCERT Solutions

(i) False. Infinitely many lines can pass through a single point. Imagine a point on a piece of paper; you can draw lines passing through it in any direction.

(ii) False. Through two distinct points, only one unique straight line can pass. This is a fundamental postulate in Euclidean geometry.

(iii) True. A terminated line, which is a line segment, can be extended indefinitely in both directions to form a straight line. This is consistent with Euclid's definition of a line.

(iv) True. If two circles are equal, it means they have the same area. The area of a circle is given by the formula $\backslash pi\; r^2$. For the areas to be equal, their radii ($r$) must also be equal.

(v) True. This statement illustrates Euclid's axiom which states that 'things which are equal to the same thing are equal to one another'. If $AB$ is equal to $PQ$, and $PQ$ is equal to $XY$, then $AB$ must be equal to $XY$.

(i) Parallel lines: Two straight lines in a plane that do not intersect at any point are called parallel lines.

Terms needing prior definition: 'Point' and 'straight line'.

Definitions:

A point is that which has no part (no dimension).

A straight line is a breadthless length (one-dimensional).

(ii) Perpendicular lines: Two lines are perpendicular if they intersect each other at a right angle (90°).

Terms needing prior definition: 'Line', 'intersection', 'right angle'. The concept of 'rotation by 90°' used in some definitions might be considered intuitive and not strictly definable from basic postulates without further assumptions.

(iii) Line segment: A line segment is a part of a line that has two distinct endpoints.

Terms needing prior definition: 'Line' and 'point'.

(iv) Radius of a circle: A radius of a circle is the line segment connecting the centre of the circle to any point on the circumference of the circle.

Terms needing prior definition: 'Circle', 'centre', and 'line segment'. A 'centre' can be defined as a point inside the circle equidistant from all points on its circumference.

(v) Square: A square is a quadrilateral where all four sides are equal in length, and all four interior angles are right angles (90°).

Terms needing prior definition: 'Quadrilateral', 'side', and 'angle'. A quadrilateral is a closed figure formed by four line segments.

Undefined terms: Postulate (i) contains the undefined term 'in between'. This term relies on our intuitive understanding of geometric space.

Consistency: Yes, these postulates are consistent. They do not contradict each other. Postulate (i) states the existence of a point between two given points, while postulate (ii) states the existence of non-collinear points. These can coexist.

Following from Euclid's postulates: These postulates do not directly follow from Euclid's five postulates. However, they are fundamental concepts that are often assumed or derived from axioms. For example, the existence of a line passing through two distinct points is a key axiom. Postulate (i) is related to the idea that a line segment contains points between its endpoints. Postulate (ii) is necessary to define a plane, as three non-collinear points uniquely determine a plane.

Explanation:

(i) If we have two distinct points A and B, Euclid's postulate implies a unique line passes through them. Any point on the line segment AB, other than A and B, is 'in between' A and B. Thus, such a point C exists.

(ii) If there were only two points in existence, they would define a line, and any other point would have to lie on this line. To have at least three points not on the same line, we need more than just two points; this is essential for defining geometric figures like triangles and planes.

Let's visualize the points on a line:

$$A\; \backslash quad\; C\; \backslash quad\; B$$

We are given that point C lies between points A and B, and the distance from A to C is equal to the distance from B to C. Mathematically, this is stated as:

$AC\; =\; BC$

We also know that the total distance from A to B is the sum of the distance from A to C and the distance from C to B. This can be written as:

$AB\; =\; AC\; +\; CB$

Since we are given that $AC\; =\; BC$, we can substitute $AC$ for $CB$ in the equation for $AB$:

$AB\; =\; AC\; +\; AC$

Combining the terms on the right side:

$AB\; =\; 2AC$

Now, to find the value of $AC$, we can divide both sides of the equation by 2:

$\backslash frac\{AB\}\{2\}\; =\; \backslash frac\{2AC\}\{2\}$

$\backslash frac\{1\}\{2\}AB\; =\; AC$

Therefore, we have proved that $AC\; =\; \backslash frac\{1\}\{2\}AB$. This shows that if a point C is exactly in the middle of A and B (meaning it bisects the segment AB), then the length of the segment AC is half the length of the entire segment AB.

To prove that every line segment has one and only one midpoint, we need to show two things: first, that a midpoint exists, and second, that there cannot be more than one midpoint.

Existence of a midpoint: From Question 4, we established that if $AC\; =\; BC$ and C is between A and B, then $AC\; =\; \backslash frac\{1\}\{2\}AB$. This implies that such a point C, which divides the line segment into two equal halves, exists.

Uniqueness of the midpoint: Let's assume that a line segment AB has two distinct midpoints, say C and D.

If C is a midpoint, then according to the result from Question 4:

$AC\; =\; \backslash frac\{1\}\{2\}AB$

If D is also a midpoint, then:

$AD\; =\; \backslash frac\{1\}\{2\}AB$

From these two equations, we can conclude that:

$AC\; =\; AD$

Since C and D are points on the line segment AB, and the distance from A to C is the same as the distance from A to D, this means that point C and point D must be the same point. They must coincide.

Therefore, our assumption that there are two distinct midpoints leads to a contradiction. This proves that a line segment cannot have more than one midpoint.

Conclusion: Every line segment has one and only one midpoint.