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Chapter 5 Introduction To Euclid's Geometry
Embark on a unique journey into the very foundations of geometry with this exploration inspired by Chapter 5, "Introduction to Euclid's Geometry," as described for the Class 9 mathematical curriculum. Unlike chapters focused heavily on calculations or constructions, this section delves into the historical development and, more importantly, the axiomatic structure upon which centuries of geometric understanding have been built. It invites students to think critically about the logical underpinnings of the subject, appreciating geometry not just as a collection of facts and formulas, but as a rigorous deductive system pioneered by the ancient Greek mathematician Euclid. The focus here shifts towards conceptual clarity and the development of precise logical reasoning skills, analyzing the 'why' behind geometric truths.
Central to Euclid's approach, and explored within the solutions framework, is the careful use of language and definitions. We examine Euclid's attempts to define fundamental concepts like a point ("that which has no part"), a line ("breadthless length"), and a surface ("that which has length and breadth only"). This leads to the crucial understanding of the difference between terms that can be reasonably defined using simpler concepts and those fundamental ideas that must be accepted as undefined terms to avoid circularity – the very building blocks upon which the entire system rests. The solutions emphasize the importance of this structure in establishing a consistent logical framework for geometry.
A cornerstone of the Euclidean system is the distinction between two types of fundamental assumptions: Axioms (or Common Notions) and Postulates. The solutions meticulously clarify this difference. Axioms are presented as self-evident truths assumed to be valid throughout all branches of mathematics, such as "Things which are equal to the same thing are equal to one another" or "The whole is greater than the part." In contrast, Postulates are assumptions specific to geometry, outlining the basic agreements about points, lines, and circles upon which geometric constructions and proofs are based. Euclid proposed five key postulates, including statements about drawing straight lines, extending line segments, drawing circles, the equality of right angles, and the famous parallel postulate.
Particular attention is often given to Euclid's Fifth Postulate, commonly known as the Parallel Postulate. This postulate is notably more complex than the others and its investigation has profound historical significance. The solutions explain its statement concerning lines intersecting a transversal and the conditions under which they meet. The significance lies in the fact that assuming different versions of this postulate leads to different, yet internally consistent, geometries (non-Euclidean geometries), a concept hinted at through discussion. Equivalent formulations, such as Playfair's Axiom ("Through a given point not on a given line, exactly one parallel line can be drawn"), might also be explored to illustrate the postulate's core idea. The exercises addressed by the solutions often involve analyzing statements for consistency with Euclid's framework and performing basic deductive proofs directly from the axioms and postulates – for instance, proving that two distinct lines cannot intersect at more than one point, thereby reinforcing logical deduction over numerical computation.
Example 1 & 2 (Before Exercise 5.1)
Example 1. If A, B and C are three points on a line, and B lies between A and C (see Fig. 5.7),, then prove that AB + BC = AC.
Answer:
Given:
A, B, and C are three points on a line.
Point B lies between points A and C.
To Prove:
$AB + BC = AC$
Proof:
We are given that B lies between A and C on a line. This means that the points A, B, and C are collinear, and B is positioned between A and C.
According to Euclid's Axiom 4, "Things which coincide with one another are equal to one another."
Consider the line segment AC. The point B divides the line segment AC into two parts: AB and BC.
The line segment AC coincides with the sum of the line segments AB and BC when they are placed adjacent to each other along the line, with B being the common endpoint.
In other words, the length of the segment AC is the sum of the lengths of the segments AB and BC because B is between A and C.
Using Euclid's Axiom 4, the length of the whole segment AC is equal to the sum of the lengths of its parts AB and BC.
Thus, we have:
$AB + BC = AC$
This is a fundamental property based on the geometric intuition that the measure of a whole is equal to the sum of the measures of its parts, which is consistent with Euclid's axioms, particularly Axiom 4 when considering magnitudes.
Statement Proved.
Example 2. Prove that an equilateral triangle can be constructed on any given line segment.
Answer:
Given:
A line segment, say AB.
To Prove:
An equilateral triangle can be constructed with AB as one of its sides.
Construction Required:
We will construct an equilateral triangle ABC on the given line segment AB using a compass.
Construction Steps:
1. With point A as the centre, draw a circle with radius equal to the length of the segment AB.
2. With point B as the centre, draw another circle with radius equal to the length of the segment AB.
3. Let the two circles intersect at a point. We can call this point C. (The circles will intersect because the radius is equal to the distance between their centres).
4. Draw the line segments AC and BC using a straight edge (ruler).
The triangle ABC is the required equilateral triangle.
Proof:
From our construction, we drew a circle with centre A and radius AB. Since point C is on this circle, the distance from A to C is equal to the radius of the circle.
AC = AB
(By Construction)
Similarly, from our construction, we drew a circle with centre B and radius AB. Since point C is on this circle, the distance from B to C is equal to the radius of the circle.
BC = AB
(By Construction)
Now we have two equalities: $AC = AB$ and $BC = AB$.
According to a basic principle (Axiom), if two things are equal to the same thing, they are equal to each other.
AC = BC
(From the above two equalities)
So, we have shown that all three sides of the triangle ABC are equal in length:
AB = AC = BC
By definition, an equilateral triangle is a triangle that has all three sides equal in length.
Since triangle ABC has all three sides equal to AB, it is an equilateral triangle constructed on the line segment AB.
Hence Proved.
Exercise 5.1
Question 1. Which 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 Fig. 5.9, if AB = PQ and PQ = XY, then AB = XY.
Answer:
Let's examine each statement and provide the reason for whether it is true or false, based on Euclidean geometry principles.
(i) Only one line can pass through a single point.
This statement is False.
Reason: Through a single point, it is possible to draw countless lines passing through it in different directions. Think of a point as the centre of a wheel; infinitely many spokes (lines) can pass through the centre.
(ii) There are an infinite number of lines which pass through two distinct points.
This statement is False.
Reason: According to Euclid's Postulate 1, "A straight line may be drawn from any one point to any other point." This guarantees the existence of at least one line. A fundamental understanding of straight lines in geometry, which is often treated as an axiom or a consequence of Postulate 1, is that there is one and only one unique straight line that passes through two distinct points.
(iii) A terminated line can be produced indefinitely on both the sides.
This statement is True.
Reason: This is a direct statement based on Euclid's Postulate 2, which says, "A terminated line can be produced indefinitely." A 'terminated line' is what we commonly call a line segment today. This postulate means that a line segment can be extended endlessly in both directions to form a complete line.
(iv) If two circles are equal, then their radii are equal.
This statement is True.
Reason: In geometry, 'equal figures' often refers to figures that are congruent, meaning they can be placed exactly on top of each other to coincide. A circle is uniquely defined by its centre and its radius. If two circles are equal (congruent), they must have the same size. The size of a circle is determined solely by the length of its radius. Therefore, if two circles are equal, their radii must be equal. This is supported by Euclid's Axiom 4, which states, "Things which coincide with one another are equal to one another."
(v) In Fig. 5.9, if AB = PQ and PQ = XY, then AB = XY.
This statement is True.
Reason: This is an application of Euclid's Axiom 1, which states, "Things which are equal to the same thing are equal to one another." Here, the length of line segment AB is equal to the length of line segment PQ. Also, the length of line segment XY is equal to the length of line segment PQ. Since both AB and XY are equal to the same line segment PQ, their lengths must be equal to each other. Therefore, AB = XY.
Question 2. Give 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
Answer:
Let's define each term and identify the basic terms that are needed before defining them, providing definitions for the preceding terms as well.
(i) Parallel lines
Definition: Two straight lines in a plane are called parallel lines if they do not meet or intersect each other, even if extended indefinitely in both directions.
Preceding terms to be defined:
Point: A point is a location that has no size or dimension. It is usually represented by a dot.
Line: A line is a straight path of points that extends infinitely in both directions without any thickness. It is defined by at least two points.
Plane: A plane is a flat surface that extends infinitely in all directions. Think of the surface of a table or a sheet of paper that goes on forever.
Intersect: When two lines cross each other or have one or more points in common, they are said to intersect.
(ii) Perpendicular lines
Definition: Two lines (or line segments or rays) are said to be perpendicular if they intersect each other and form a right angle at the point of intersection.
Preceding terms to be defined:
Line: A line is a straight path of points that extends infinitely in both directions without any thickness. It is defined by at least two points.
Intersect: When two lines cross each other or have one or more points in common, they are said to intersect.
Angle: An angle is formed when two lines or rays meet at a common point. The point where they meet is called the vertex.
Right angle: A right angle is a special type of angle that measures exactly $90^\circ$. It looks like the corner of a square or a rectangle.
(iii) Line segment
Definition: A line segment is a part of a line that is bounded by two distinct endpoints. It has a definite length.
Preceding terms to be defined:
Point: A point is a location that has no size or dimension. It is usually represented by a dot.
Line: A line is a straight path of points that extends infinitely in both directions without any thickness. It is defined by at least two points.
Endpoint: A point that marks the beginning or the end of a line segment or a ray.
(iv) Radius of a circle
Definition: The radius of a circle is the distance from the centre of the circle to any point on the circle's boundary (circumference).
Preceding terms to be defined:
Point: A point is a location that has no size or dimension. It is usually represented by a dot.
Circle: A circle is a set of all points in a plane that are at the same fixed distance from a fixed point within it.
Centre (of a circle): The fixed point inside the circle from which all points on the circle are equally distant.
Distance: The distance between two points is the length of the straight line segment joining those points.
Length: Length is a measure of how long a line segment is. It is often considered a basic concept.
Line segment: A line segment is a part of a line that is bounded by two distinct endpoints. It has a definite length.
Line: A line is a straight path of points that extends infinitely in both directions without any thickness. It is defined by at least two points.
Endpoint: A point that marks the beginning or the end of a line segment or a ray.
Plane: A plane is a flat surface that extends infinitely in all directions. Think of the surface of a table or a sheet of paper that goes on forever.
(v) Square
Definition: A square is a quadrilateral (a four-sided figure) where all four sides are equal in length and all four interior angles are right angles.
Preceding terms to be defined:
Point: A point is a location that has no size or dimension. It is usually represented by a dot.
Line segment: A line segment is a part of a line that is bounded by two distinct endpoints. It has a definite length.
Line: A line is a straight path of points that extends infinitely in both directions without any thickness. It is defined by at least two points.
Endpoint: A point that marks the beginning or the end of a line segment or a ray.
Side: A side of a geometric figure is one of the line segments that form its boundary.
Quadrilateral: A quadrilateral is a closed plane figure formed by four line segments joined endpoint to endpoint.
Plane: A plane is a flat surface that extends infinitely in all directions. Think of the surface of a table or a sheet of paper that goes on forever.
Length: Length is a measure of how long a line segment is. It is often considered a basic concept.
Equal (in length): Two line segments are equal in length if they have the same measure.
Angle: An angle is formed when two lines or rays meet at a common point. The point where they meet is called the vertex.
Right angle: A right angle is a special type of angle that measures exactly $90^\circ$. It looks like the corner of a square or a rectangle.
Question 3. Consider 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.
Answer:
Let's analyze the two given postulates:
Postulate (i): Given any two distinct points A and B, there exists a third point C which is in between A and B.
Postulate (ii): There exist at least three points that are not on the same line.
Do these postulates contain any undefined terms?
Yes, these postulates contain undefined terms. The terms that are undefined in these postulates, and typically in geometry systems (including Euclid's, where they are implicitly understood rather than formally defined exhaustively), include:
- Point: What exactly constitutes a "point"?
- Line: What exactly constitutes a "line"?
- In between: What does it mean for one point to be "in between" two other points on a line?
- Distinct points: What makes two points "distinct"?
- Are not on the same line: This uses the concept of a line and points lying on a line, which relies on the undefined term "line".
Euclid himself used terms like point, line, and plane without providing rigorous definitions, treating them as intuitive concepts. The concept of "in between" is also often taken as an undefined or foundational concept in axiomatic geometry.
Are these postulates consistent?
Yes, these postulates are consistent. Two postulates are consistent if it is not possible to deduce a contradiction from them. We can visualize a scenario where both postulates hold true simultaneously.
Postulate (i) suggests that a line is continuous, with points existing between any two given points. For example, on a number line, between any two distinct numbers $a$ and $b$, there is always a number $\frac{a+b}{2}$ which is between them. This can be extended indefinitely, implying infinitely many points on a line.
Postulate (ii) states that there are points not all lying on a single line. This is true in Euclidean geometry (and many other geometries). For example, the vertices of a triangle (three points not on the same line) satisfy this condition. If all points were on the same line, geometry would only exist in one dimension.
Both postulates describe properties of the geometric world we commonly understand. Postulate (i) relates to the density or continuity of points on a line, while Postulate (ii) guarantees that geometry is not restricted to a single line, allowing for two-dimensional (and higher) figures.
There is no logical contradiction between the idea that there are points between any two points on a line and the idea that not all points lie on the same line.
Do they follow from Euclid’s postulates?
Postulate (i) does not directly follow from Euclid's postulates, but it is consistent with the system based on them and subsequent developments in understanding the nature of lines and points. Euclid's postulates imply that a line can be produced indefinitely, which suggests continuity, but postulate (i) explicitly states the existence of a point between any two points. This concept of "betweenness" is fundamental and might be considered an additional axiom needed for a complete axiomatic system of geometry.
Postulate (ii) also does not directly follow as a logical consequence from Euclid's postulates. Euclid's postulates describe properties of points and lines and how they relate, but they don't explicitly guarantee the existence of points *not* on a given line or the existence of non-collinear points. However, Euclid's constructions (like constructing an equilateral triangle on a line segment, as in Example 2) implicitly rely on the existence of points outside a given line segment. The existence of figures like triangles (which require three non-collinear points) in Euclidean geometry demonstrates that postulate (ii) is consistent with and necessary for developing Euclidean geometry beyond a single line.
In modern axiomatic treatments of geometry, postulates similar to (i) and (ii) are often included as part of the axioms (e.g., axioms of order or existence) that, together with others, form the basis of Euclidean geometry.
Explanation:
The postulates contain undefined terms like 'point', 'line', and 'in between'.
They are consistent because they describe possible situations in geometry without contradiction. Postulate (i) suggests the dense nature of points on a line, while postulate (ii) ensures the existence of a plane (or higher dimensions) by guaranteeing that not all points are collinear.
They do not directly follow from Euclid's postulates as listed in The Elements, but they represent fundamental properties of Euclidean space that are either implicitly assumed or require additional axioms in a rigorous axiomatic system.
Question 4. If a point C lies between two points A and B such that AC = BC, then prove that AC = $\frac{1}{2}$AB. Explain by drawing the figure.
Answer:
Given:
Points A and B are two distinct points.
Point C lies between A and B on the line segment AB.
AC = BC
To Prove:
$AC = \frac{1}{2}AB$
Figure:
Let's draw a line segment AB with point C between A and B.
Proof:
Since point C lies between points A and B on the line segment AB, the length of the entire segment AB is equal to the sum of the lengths of the segments AC and CB.
This is a basic property of line segments (sometimes considered an axiom or postulate).
AC + CB = AB
... (i)
We are given that the length of segment AC is equal to the length of segment BC.
AC = BC
(Given)
Since C lies between A and B, CB is the same segment as BC, so their lengths are equal: CB = BC.
Thus, we can use the given condition AC = BC (or AC = CB) and substitute AC for CB in equation (i).
Substituting BC with AC in equation (i), we get:
$AC + AC = AB$
Combine the two terms on the left side:
$2 \cdot AC = AB$
Now, to find the value of AC, we divide both sides of the equation by 2.
$\frac{2 \cdot AC}{2} = \frac{AB}{2}$
This simplifies to:
$AC = \frac{1}{2}AB$
Thus, we have proved that if a point C lies between two points A and B such that AC = BC, then $AC = \frac{1}{2}AB$.
Explanation with the figure:
The figure shows that points A, C, and B are on a straight line in that order. This means the total length from A to B is the sum of the lengths from A to C and from C to B.
Length of AB = Length of AC + Length of CB
We are told that $AC = BC$. This means that point C is exactly in the middle of the line segment AB. It divides the segment into two parts of equal length.
If the two parts are equal (AC and BC) and they add up to the whole (AB), then each part must be exactly half of the whole length.
Since $AC = BC$, we can think of AB as being made up of two segments, both with the same length as AC.
$AB = AC + AC$
$AB = 2 \times AC$
Dividing the total length AB into two equal parts means that the length of each part (AC or BC) is half of the total length AB.
$AC = \frac{1}{2}AB$
This makes sense because C is the midpoint of AB.
Hence Proved.
Question 5. In 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.
Answer:
Given:
A line segment AB.
A point C is called a midpoint of AB if C lies between A and B and AC = BC.
To Prove:
Every line segment has one and only one midpoint.
Proof:
We need to prove two things: (1) Existence of a midpoint, and (2) Uniqueness of a midpoint.
(1) Existence: Every line segment has at least one midpoint.
Consider the line segment AB. It has a definite length. By geometric construction (which is possible based on Euclid's postulates, e.g., bisecting a line segment), or by considering the real number line model where point A corresponds to number $a$ and point B corresponds to number $b$, there exists a unique point C on the line segment AB such that the distance AC is exactly half the distance of AB.
Let's assume the length of AB is $L$. We can find a point C on the segment AB such that $AC = \frac{1}{2} L$. Since C is on the segment AB and $AC = \frac{1}{2} AB$, the remaining length CB must be:
$CB = AB - AC = L - \frac{1}{2} L = \frac{1}{2} L$
So, we have $AC = \frac{1}{2} L$ and $CB = \frac{1}{2} L$, which means $AC = CB$. Also, since AC is less than AB (as AC is half of AB) and C is on the line containing AB, C must lie between A and B.
Thus, there exists a point C between A and B such that AC = BC, which means there is at least one midpoint.
(2) Uniqueness: Every line segment has at most one midpoint.
Assume for the sake of contradiction that the line segment AB has two distinct midpoints, say C and D.
If C is a midpoint of AB, then C lies between A and B, and AC = BC.
From the result of Question 4, if C is between A and B such that AC = BC, then $AC = \frac{1}{2} AB$.
If D is a midpoint of AB, then D lies between A and B, and AD = BD.
Similarly, from the result of Question 4, if D is between A and B such that AD = BD, then $AD = \frac{1}{2} AB$.
So, we have:
$AC = \frac{1}{2} AB$
... (ii)
$AD = \frac{1}{2} AB$
... (iii)
From (ii) and (iii), by Euclid's Axiom 1 ("Things which are equal to the same thing are equal to one another"), we have:
AC = AD
Now, since C and D are assumed to be two distinct points, and both lie on the line segment AB (between A and B), one of them must lie between the other and A (or B). Without loss of generality, let's assume C lies between A and D.
If C lies between A and D, then the length of the segment AD is the sum of the lengths of segments AC and CD:
$AD = AC + CD$
Since C and D are distinct points, the length of the segment CD must be greater than 0 ($CD > 0$).
So, $AD = AC + CD > AC$.
This implies $AD > AC$.
However, we concluded earlier that $AC = AD$. This contradicts our finding that $AD > AC$.
The only way for this contradiction to be avoided is if our initial assumption that C and D are distinct points is false. Therefore, C and D must be the same point.
This proves that there cannot be two distinct midpoints; a line segment can have at most one midpoint.
From parts (1) and (2), we conclude that every line segment has one and only one midpoint.
Statement Proved.
Question 6. In Fig. 5.10, if AC = BD, then prove that AB = CD.
Answer:
Given:
Points A, B, C, D are on a line as shown in Fig. 5.10.
AC = BD
To Prove:
AB = CD
Proof:
From the figure, we can see that point B lies between A and C. Therefore, according to the segment addition property (based on Euclid's axioms), the length of the whole segment AC is the sum of the lengths of the parts AB and BC.
AC = AB + BC
... (i)
Similarly, point C lies between B and D. Therefore, the length of the whole segment BD is the sum of the lengths of the parts BC and CD.
BD = BC + CD
... (ii)
We are given that AC = BD.
AC = BD
(Given)
Substitute the expressions for AC and BD from equations (i) and (ii) into the given equality:
$AB + BC = BC + CD$
According to Euclid's Axiom 3, "If equals be subtracted from equals, the remainders are equal."
Here, AB + BC and BC + CD are equal. We can subtract BC (which is equal to itself) from both sides of the equation.
$AB + BC - BC = BC + CD - BC$
$AB + 0 = 0 + CD$
$AB = CD$
Thus, we have proved that if AC = BD, then AB = CD.
Statement Proved.
Question 7. Why is Axiom 5, in the list of Euclid’s axioms, considered a ‘universal truth’? (Note that the question is not about the fifth postulate.)
Answer:
Euclid's Common Notions (Axioms) are statements that were considered self-evident and applicable to all fields of mathematics, not just geometry. Euclid's Fifth Axiom (or Common Notion 5) is typically stated as:
"The whole is greater than the part."
This axiom is considered a 'universal truth' because its validity is intuitively obvious and does not require any proof. It holds true in various contexts, not just geometrical ones. It is a fundamental principle about magnitudes and quantities.
Here's why it is considered a universal truth:
1. Self-Evident Nature: The statement "The whole is greater than the part" is intuitively understood by everyone. If you have a whole object or quantity and take a part of it, the original whole is clearly larger than the piece you took away.
2. Applicability Beyond Geometry: This axiom applies universally to any kind of magnitude or quantity, whether it's length, area, volume, weight, numbers, etc.
- In geometry: The length of a line segment is greater than the length of any smaller segment within it. The area of a figure is greater than the area of any part of that figure.
- In arithmetic: A number (the whole) is greater than any of its proper parts. For example, 5 is greater than 3.
- In other contexts: The weight of a whole apple is greater than the weight of a slice of that apple. The total duration of an event is greater than the duration of any phase of that event.
3. Basis for Comparisons: It forms a fundamental basis for understanding and comparing sizes or quantities. It's a foundational idea in the concept of 'greater than'.
Unlike Euclid's Postulates (which are specific assumptions about geometric constructions and lines), the Axioms (Common Notions) were intended to be fundamental truths accepted without question across different mathematical domains. Axiom 5, in particular, is so universally applicable and evident that it is considered a core, undeniable truth about quantities and magnitudes.