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The exercise establishes the fundamental basis of Euclidean geometry through its presentation. This turns toward knowing Euclid's method of constructing logical geometry with his definitions and axioms and postulates. The programmed approach created fundamentals to produce advanced geometrical theorems and constructions by tracing facts down from essential starting points thus establishing the core structure of classical geometry.
Through this exercise students learn to differentiate between axioms and postulates by analyzing their importance as well as their application process for creating mathematical logic systems. Students can better comprehend complex abstract geometrical concepts through these guided NCERT Solutions which function perfectly with regular NCERT Books for constructing a firm foundation in geometry.
Q1 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:
i) FALSE
Because, the single point allows an infinite number of parallel lines to be drawn through it. As shown in the diagram below
ii) FALSE
Because, one line exists between two distinct points because of fundamental mathematical principles. As shown in the diagram below
iii) TRUE
Because, the production of a terminated line remains possible without limits from both sides. As shown in the diagram below
iv) TRUE
Because, when two circles match in size then both their central points together with their outer boundaries align perfectly while their radii become equal with each other.
v) TRUE
By Euclid’s first axiom things which are equal to the same thing, are equal to one another.
Answer:
Yes, there are other terms that are needed to be defined first which are:
Plane: An infinite flat two-dimensional surface defines what a plane represents
Point: Points represent small dots which hold no dimensions like length, breadth or height and exist only to indicate position.
Line: The definition of a line describes an infinite straight one-dimensional shape without thickness which continues indefinitely forward and backward.
i) Parallel line:-
if two lines in a plane never cross one another and their perpendicular distance is always constant. Two lines are thus referred to as parallel lines.
Answer:
Yes, there are other terms that are needed to be defined first which are:
Plane: An infinite flat two-dimensional surface defines what a plane represents
Point: Points represent small dots which hold no dimensions like length, breadth or height and exist only to indicate position.
Line: The definition of a line describes an infinite straight one-dimensional shape without thickness which continues indefinitely forward and backward.
ii) perpendicular line:-
If two lines intersect with each other and make a right angle at the point of intersection. Then, two lines are called perpendicular lines.
Answer:
Yes, there are other terms that are needed to be defined first which are:
Plane: An infinite flat two-dimensional surface defines what a plane represents
Point: Points represent small dots which hold no dimensions like length, breadth or height and exist only to indicate position.
Line: The definition of a line describes an infinite straight one-dimensional shape without thickness which continues indefinitely forward and backward.
iii) line segment:-
A line segment is a straight line having two end points that have a fixed length and cannot be stretched further.
Answer:
Yes, there are other terms that are needed to be defined first which are:
Plane: An infinite flat two-dimensional surface defines what a plane represents
Point: Points represent small dots which hold no dimensions like length, breadth or height and exist only to indicate position.
Line: The definition of a line describes an infinite straight one-dimensional shape without thickness which continues indefinitely forward and backward.
iv) Radius of the circle: -
The radius of a circle is the separation between the circle's center and any point on its circumference.
Q2 (v) 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? v) square
Answer:
Yes, there are other terms that are needed to be defined first which are:
Plane: An infinite flat two-dimensional surface defines what a plane represents
Point: Points represent small dots which hold no dimensions like length, breadth or height and exist only to indicate position.
Line: The definition of a line describes an infinite straight one-dimensional shape without thickness which continues indefinitely forward and backward.
v) Square:-
Any quadrilateral with equal four sides and right angles at all interior angles is called a square.
Q3 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:
The following postulates contain a number of undefined terms:
1) No information regarding the plane is provided, including whether or not the points are in the same plane.
2) A plane has an endless number of points. However, it is unclear from this point C's location whether it is on the line segment that joins AB or not.
Yes, these postulates hold true and consistent in the following two scenarios:
(i) Point C is located on the line segment that connects A and B.
(ii) The line segment connecting A and B does not contain Point C.
No, Euclid's postulates don't support them. They adhere to the principles.
Answer:
Given, AC = BC
Now, draw the figure as below:
In the figure, AB coincides with AC + BC.
Also, Euclid’s Axiom (4) says that things which coincide with one another are equal to one another. Hence, it is deduced that AC + BC = AB
So from this we can conclude; 2AC = AB
Therefore,
Hence proved.
Answer:
Let's consider, there are two midpoints C and D of a line segment A and B.
Now, If C is the midpoint then, AC = BC
And, this concludes, AB coincides with AC + BC.
Also, Euclid’s Axiom (4) says that things which coincide with one another are equal to one another. So, it can be deduced that AC + BC = AB
Therefore, we can say that
2AC = AB -(i)
Similarly, in case of point D.
If D is the midpoint then, AD = BD
And, when considering the figure we can conclude, AB coincides with AD + BD.
Also, Euclid’s Axiom (4) says that things which coincide with one another are equal to one another. So, it can be deduced that AD + BD = AB
From this, we can say that
2AD = AB -(ii)
Now,
From equation (i) and (ii) we get
AD = AC
and this is only possible when C and D are the same points
Hence, our assumption is wrong and there is only one midpoint of line segment AB.
Q6 In Fig. 5.10, if AC = BD, then prove that AB = CD.
Answer:
From the figure given,
We can say, AC = AB + BC and BD = BC + CD
It is given that AC = BD
Therefore, we can write:
AB + BC = BC + CD
Now, According to Euclid's axiom, when equals are subtracted from equals, the remainders are also equal. Subtracting BC from both sides.
We get: AB + BC - BC = BC + CD - BC
AB = CD
Hence proved
Answer:
Axiom 5 states that the whole is greater than the part.
Let's Understand with the help of an example:
Let A = x + y + z (where A , x , y , z all are positive numbers)
Now, we can clearly see that A > x , A > y , A > z
Hence, by this we can say that the whole (A) is greater than the parts. (x , y , z)
Other real life examples:
Also see-
Euclid's geometry is defined as the study of geometry based on undefined notions such as points, lines, and planes of flat spaces, according to NCERT solutions for Class 9 Maths chapter 5 exercise 5.1
Euclid’s geometry was introduced by Euclid.
Euclid is the Father of Geometry.
The lines that intersect each other in a plane at right angles are said to be perpendicular to each other, according to NCERT solutions for Class 9 Maths chapter 5 exercise 5.1
The number of axioms of Euclid’s geometry is 7.
The correct Euclid’s axiom that illustrates the relative ages of Rani and Sudha is the first axiom. According to the first axiom, Things that are equal to the same thing are equal to one another.
There are an endless number of lines that can pass through a given spot.
Things that are similar to one another are referred to as equal.
Reason: According to axiom 4, things that are at the same place at the same time are equal.
A straight line segment can be drawn for any two given points as stated in the form of a postulate
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