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

(i) False. Infinitely many lines can pass through a single point. Imagine a point as the center; you can draw lines radiating outwards in every possible direction.

(ii) False. Through two distinct points, there exists exactly one unique line that passes through them. This is a fundamental postulate in geometry.

(iii) True. A terminated line is a line segment. According to Euclid's postulates, a line segment can be extended indefinitely on both sides to form a straight 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 $\backslash pi\; r^2$. If the areas are equal, then $\backslash pi\; r\_1^2\; =\; \backslash pi\; r\_2^2$, which implies $r\_1^2\; =\; r\_2^2$. Since radii must be positive, $r\_1\; =\; r\_2$. Thus, their radii are equal.

(v) True. This statement illustrates Euclid's axiom of equality, which states that if two quantities are equal to a third quantity separately, then they are equal to each other. Here, if $AB$ is equal to $PQ$, and $PQ$ is also 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.

A line is a breadthless length. A straight line is a line which lies evenly with the points on itself.

(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'. Euclid's definition often relies on intuition or prior concepts like rotation, which themselves need careful definition. For instance, 'rotation through 90°' is not explicitly defined but assumed.

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

Terms needing prior definition: 'Line' and 'point', which have been defined above.

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

Terms needing prior definition: 'Centre' (a point inside the circle equidistant from all points on the circumference), 'circle', and 'point'.

(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' (a polygon with four sides), 'side', and 'angle'. These are generally assumed to be understood from basic geometry.

Undefined Terms:

Postulate (i) contains the undefined term 'in between'. While we intuitively understand what it means for a point to be between two other points on a line, Euclid's system requires precise definitions or axioms to handle such concepts rigorously.

Consistency:

Yes, these postulates are consistent. They do not contradict each other. Postulate (i) states the existence of a point between two given points, and 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 consistent with Euclid's axioms and the general understanding of geometry. For example:

1. Postulate (i) is related to the axiom that states: Given two distinct points, there is a unique line that passes through them. If we have points A and B, the line AB contains infinitely many points between A and B.

2. Postulate (ii) is also fundamental. If there were only two points, they would define a single line, and any third point would have to lie on that line. The existence of at least three non-collinear points is necessary for defining shapes like triangles.

Let's visualize the points on a line:

$\backslash stackrel\{\backslash leftarrow\}\{A\}$------------------$C$------------------$\backslash stackrel\{\backslash leftarrow\}\{B\}$

We are given that point C lies between points A and B, and that the length of the segment AC is equal to the length of the segment BC. Mathematically, this is stated as $AC\; =\; BC$.

We also know that the entire length of the line segment AB is the sum of the lengths of the segments AC and CB, because C lies between A and B. So, $AB\; =\; AC\; +\; CB$.

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

This gives us: $AB\; =\; AC\; +\; AC$

Combining the terms on the right side, we get: $AB\; =\; 2AC$

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

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

This simplifies to: $AC\; =\; \backslash frac\{1\}\{2\}AB$

Thus, we have proved that if a point C lies between A and B such that $AC\; =\; BC$, then $AC$ is half the length of $AB$.

To prove that every line segment has one and only one midpoint, we need to show two things:

1. That a midpoint exists.

2. That there cannot be more than one midpoint.

Existence of a Midpoint:

From Question 4, we established that if a point C lies between A and B such that AC = BC, then AC = \frac{1}{2}AB. This shows that such a point C exists and satisfies the condition of being a midpoint.

Uniqueness of the Midpoint:

Let's assume, for the sake of contradiction, that a line segment AB has two distinct midpoints, say C and D.

If C is a midpoint of AB, then from Question 4, we have:

AC = \frac{1}{2}AB

If D is also a midpoint of AB, then we have:

AD = \frac{1}{2}AB

From these two equations, we can conclude that AC = AD (since both are equal to \frac{1}{2}AB).

Now, consider the line segment AB. If C and D are distinct points on this segment, and AC = AD, this implies that the distance from A to C is the same as the distance from A to D. For this to be true, the points C and D must coincide. In other words, C and D must be the same point.

This contradicts our initial assumption that C and D are distinct midpoints. Therefore, a line segment cannot have more than one midpoint.

Conclusion:

Every line segment has one and only one midpoint.