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All questions of Introduction to Euclid`s Geometry for Class 9 Exam
Euclid was a teacher of mathematics at ?
- a) Egypt
- b)Greece
- c)Babylonia
- d)Rome
Correct answer is option 'A'. Can you explain this answer?
Euclid was a teacher of mathematics at ?
a)
Egypt
b)
Greece
c)
Babylonia
d)
Rome
|
Tanvi Singh answered |
Euclid belongs to Greece.
Euclid was a Greek mathematician who lived during the 3rd century BCE. He is often referred to as the "Father of Geometry" due to his significant contributions to the field. Euclid's work, known as "Elements," is a comprehensive compilation of mathematical theorems and proofs that laid the foundation for geometry as a formal discipline.
1. Early Life:
Euclid's exact place and date of birth are unknown, but he is believed to have been born in the Greek city of Alexandria, which was a center of learning and intellectual activity during that time. Alexandria was located in present-day Egypt, but it was under Greek rule and had a strong Greek influence.
2. Contributions to Mathematics:
Euclid's most famous work, "Elements," consists of thirteen volumes and covers a wide range of mathematical topics, including geometry, number theory, and algebra. In "Elements," Euclid presents a systematic and logical approach to geometry, starting with basic definitions and axioms and then building upon them to prove various theorems.
3. Euclidean Geometry:
Euclid's geometry, also known as Euclidean geometry, is based on five fundamental postulates, or axioms, which are accepted without proof. These postulates form the basis for all the theorems and proofs in "Elements." Euclidean geometry deals with points, lines, angles, and shapes in two and three dimensions.
4. Influence and Legacy:
Euclid's work had a profound impact on the development of mathematics and science. "Elements" became the standard textbook on geometry for over two thousand years and was studied by scholars and mathematicians across different cultures and civilizations. Euclidean geometry formed the basis for many practical applications, such as architecture, engineering, and navigation.
In conclusion, Euclid was a Greek mathematician who contributed significantly to the field of geometry. His work, "Elements," laid the foundation for geometry as a formal discipline and had a lasting impact on the development of mathematics.
Euclid was a Greek mathematician who lived during the 3rd century BCE. He is often referred to as the "Father of Geometry" due to his significant contributions to the field. Euclid's work, known as "Elements," is a comprehensive compilation of mathematical theorems and proofs that laid the foundation for geometry as a formal discipline.
1. Early Life:
Euclid's exact place and date of birth are unknown, but he is believed to have been born in the Greek city of Alexandria, which was a center of learning and intellectual activity during that time. Alexandria was located in present-day Egypt, but it was under Greek rule and had a strong Greek influence.
2. Contributions to Mathematics:
Euclid's most famous work, "Elements," consists of thirteen volumes and covers a wide range of mathematical topics, including geometry, number theory, and algebra. In "Elements," Euclid presents a systematic and logical approach to geometry, starting with basic definitions and axioms and then building upon them to prove various theorems.
3. Euclidean Geometry:
Euclid's geometry, also known as Euclidean geometry, is based on five fundamental postulates, or axioms, which are accepted without proof. These postulates form the basis for all the theorems and proofs in "Elements." Euclidean geometry deals with points, lines, angles, and shapes in two and three dimensions.
4. Influence and Legacy:
Euclid's work had a profound impact on the development of mathematics and science. "Elements" became the standard textbook on geometry for over two thousand years and was studied by scholars and mathematicians across different cultures and civilizations. Euclidean geometry formed the basis for many practical applications, such as architecture, engineering, and navigation.
In conclusion, Euclid was a Greek mathematician who contributed significantly to the field of geometry. His work, "Elements," laid the foundation for geometry as a formal discipline and had a lasting impact on the development of mathematics.
The number of dimensions, a surface has- a)1
- b)0
- c)2
- d)3
Correct answer is option 'C'. Can you explain this answer?
The number of dimensions, a surface has
a)
1
b)
0
c)
2
d)
3
|
|
Arun Yadav answered |
A solid has 3 dimensions, the surface has 2, the line has 1 and the point is dimensionless.
#Follow
#Follow
If the point P lies in between M and N and C is midpoint of MP then:
- a)MP + CP = MN
- b)MP + CP + MN
- c)MC + CN = MN
- d)CP + CN = MN
Correct answer is option 'C'. Can you explain this answer?
If the point P lies in between M and N and C is midpoint of MP then:
a)
MP + CP = MN
b)
MP + CP + MN
c)
MC + CN = MN
d)
CP + CN = MN
|
Geetika Menon answered |
Explanation:
When a point P lies between two points M and N, we can say that M-P-N.
Given that C is the midpoint of MP.
So, we can say that MC = CP.
Also, as we know that C is the midpoint of MP, we can say that MP = 2*CP.
Now, we can write MN as:
MN = MP + PN
Substituting the value of MP in the above equation, we get:
MN = 2*CP + PN
Now, we can write PN as:
PN = CN - CP
Substituting the value of PN in the above equation, we get:
MN = 2*CP + CN - CP
Simplifying the above equation, we get:
MN = CP + CN
Therefore, we can say that MC * CN = MN.
Hence, option 'C' is the correct answer.
When a point P lies between two points M and N, we can say that M-P-N.
Given that C is the midpoint of MP.
So, we can say that MC = CP.
Also, as we know that C is the midpoint of MP, we can say that MP = 2*CP.
Now, we can write MN as:
MN = MP + PN
Substituting the value of MP in the above equation, we get:
MN = 2*CP + PN
Now, we can write PN as:
PN = CN - CP
Substituting the value of PN in the above equation, we get:
MN = 2*CP + CN - CP
Simplifying the above equation, we get:
MN = CP + CN
Therefore, we can say that MC * CN = MN.
Hence, option 'C' is the correct answer.
Axiom or pastulates are- a)Conclusions
- b)Reasons
- c)Assumptions
- d)Questions
Correct answer is option 'C'. Can you explain this answer?
Axiom or pastulates are
a)
Conclusions
b)
Reasons
c)
Assumptions
d)
Questions
|
Arzoo Sharma answered |
Axioms and postulates are assumption that are taken bu euclid to prove several theorems , axioms and postulates can not be proved as they are universal truth .
The number of line segments determined by three collinear points is:a)Twob)Threec)Fourd)Only oneCorrect answer is option 'D'. Can you explain this answer?
|
Arvind Singh answered |
if the points are collinear then only 1 line can pass through 3 points as colinear mean the points which are on same line.
If B lies on line AC and points A, B and C are distinct such that, AB + BC = AC, then- a)AB < AC
- b)AB > AC
- c)AB = AC
- d)None of these
Correct answer is option 'A'. Can you explain this answer?
If B lies on line AC and points A, B and C are distinct such that, AB + BC = AC, then
a)
AB < AC
b)
AB > AC
c)
AB = AC
d)
None of these
|
Jasneet Kaur answered |
Because AB is a part of AC and by Euclid fifth axiom which states that the whole is greater than the part ::AB < ac="" />
Can two intersecting lines be parallel to a common line?- a)sometimes
- b)Maybe
- c)Yes
- d)No
Correct answer is option 'D'. Can you explain this answer?
Can two intersecting lines be parallel to a common line?
a)
sometimes
b)
Maybe
c)
Yes
d)
No
|
Sabrina Singh answered |
Explanation:
Parallel lines are lines that never intersect, while intersecting lines are lines that cross each other at a point. Therefore, two intersecting lines cannot be parallel to a common line. This can be explained in the following ways:
Proof:
1. Definition of Parallel Lines: Two lines are said to be parallel if they lie in the same plane and do not intersect.
2. Definition of Intersecting Lines: Two lines are said to be intersecting if they meet or cross each other at a point.
3. If two lines are parallel, they can never intersect. This is because the definition of parallel lines states that they do not intersect.
4. If two lines intersect, they can never be parallel. This is because the definition of parallel lines states that they do not intersect.
5. Therefore, two intersecting lines cannot be parallel to a common line. This is because if the two lines intersect, they cannot be parallel, and if they are parallel, they cannot intersect.
Conclusion:
In conclusion, two intersecting lines cannot be parallel to a common line. This is because the definition of parallel lines states that they do not intersect, while the definition of intersecting lines states that they do intersect. Therefore, the correct answer to the question is option 'D', which is 'No'.
Parallel lines are lines that never intersect, while intersecting lines are lines that cross each other at a point. Therefore, two intersecting lines cannot be parallel to a common line. This can be explained in the following ways:
Proof:
1. Definition of Parallel Lines: Two lines are said to be parallel if they lie in the same plane and do not intersect.
2. Definition of Intersecting Lines: Two lines are said to be intersecting if they meet or cross each other at a point.
3. If two lines are parallel, they can never intersect. This is because the definition of parallel lines states that they do not intersect.
4. If two lines intersect, they can never be parallel. This is because the definition of parallel lines states that they do not intersect.
5. Therefore, two intersecting lines cannot be parallel to a common line. This is because if the two lines intersect, they cannot be parallel, and if they are parallel, they cannot intersect.
Conclusion:
In conclusion, two intersecting lines cannot be parallel to a common line. This is because the definition of parallel lines states that they do not intersect, while the definition of intersecting lines states that they do intersect. Therefore, the correct answer to the question is option 'D', which is 'No'.
Theorems are statements which are proved using definitions, _________, previously proved statements and deductive reasoning.- a)Definitions
- b)Axioms
- c)Theorems
- d)Statements
Correct answer is option 'B'. Can you explain this answer?
Theorems are statements which are proved using definitions, _________, previously proved statements and deductive reasoning.
a)
Definitions
b)
Axioms
c)
Theorems
d)
Statements
|
Pranab Datta answered |
Explanation:
The correct answer is option B - Axioms.
Definitions:
Definitions are the statements that explain the meaning of a concept or term. They provide clarity and understanding by defining the key elements of a subject.
Theorems:
Theorems are statements that are proven using definitions, axioms, previously proved statements, and deductive reasoning. They are the logical conclusions derived from a combination of these elements.
Axioms:
Axioms, also known as postulates, are statements that are accepted as true without proof. They serve as the foundation upon which all other theorems and proofs are built. Axioms are self-evident truths or statements that are assumed to be true based on common knowledge or intuition.
Role of Axioms:
Axioms provide the starting point for mathematical reasoning and form the basis of logical deductions. They are used to derive new theorems and proofs through a systematic process of deductive reasoning. Axioms are essential in mathematics as they establish the fundamental principles and rules that govern the subject.
Process of Proving Theorems:
To prove a theorem, one must start with the given definitions, axioms, and previously proven statements. These serve as the building blocks for the proof, allowing the mathematician to logically deduce new conclusions.
The process of proving a theorem involves:
1. Understanding the definitions: The definitions of the terms involved in the theorem must be clearly understood to proceed with the proof.
2. Applying axioms: The axioms are used as the initial assumptions or logical principles from which the proof starts.
3. Using previously proved statements: Previously proven statements, also known as lemmas or corollaries, are incorporated into the proof to establish a chain of reasoning.
4. Deductive reasoning: Deductive reasoning is used to make logical connections between the given information, definitions, axioms, and previously proven statements. Each step of the proof must be justified using deductive reasoning.
By following these steps, mathematicians can construct a logical and rigorous proof that establishes the truth of a theorem. Theorems are fundamental to mathematics as they provide a solid framework for building further knowledge and understanding in the subject.
The correct answer is option B - Axioms.
Definitions:
Definitions are the statements that explain the meaning of a concept or term. They provide clarity and understanding by defining the key elements of a subject.
Theorems:
Theorems are statements that are proven using definitions, axioms, previously proved statements, and deductive reasoning. They are the logical conclusions derived from a combination of these elements.
Axioms:
Axioms, also known as postulates, are statements that are accepted as true without proof. They serve as the foundation upon which all other theorems and proofs are built. Axioms are self-evident truths or statements that are assumed to be true based on common knowledge or intuition.
Role of Axioms:
Axioms provide the starting point for mathematical reasoning and form the basis of logical deductions. They are used to derive new theorems and proofs through a systematic process of deductive reasoning. Axioms are essential in mathematics as they establish the fundamental principles and rules that govern the subject.
Process of Proving Theorems:
To prove a theorem, one must start with the given definitions, axioms, and previously proven statements. These serve as the building blocks for the proof, allowing the mathematician to logically deduce new conclusions.
The process of proving a theorem involves:
1. Understanding the definitions: The definitions of the terms involved in the theorem must be clearly understood to proceed with the proof.
2. Applying axioms: The axioms are used as the initial assumptions or logical principles from which the proof starts.
3. Using previously proved statements: Previously proven statements, also known as lemmas or corollaries, are incorporated into the proof to establish a chain of reasoning.
4. Deductive reasoning: Deductive reasoning is used to make logical connections between the given information, definitions, axioms, and previously proven statements. Each step of the proof must be justified using deductive reasoning.
By following these steps, mathematicians can construct a logical and rigorous proof that establishes the truth of a theorem. Theorems are fundamental to mathematics as they provide a solid framework for building further knowledge and understanding in the subject.
Which among these is the relation between whole and the part?- a)W < P
- b)W > P
- c)W = P
- d)None of these
Correct answer is option 'B'. Can you explain this answer?
Which among these is the relation between whole and the part?
a)
W < P
b)
W > P
c)
W = P
d)
None of these
|
Sonal Sinha answered |
To understand relation between whole and a part we can take 2 and it's part 1/2.2>1/2.
If a > b and b > c, then,- a)a > c
- b)a < c
- c)a = c
- d)None of these
Correct answer is option 'A'. Can you explain this answer?
If a > b and b > c, then,
a)
a > c
b)
a < c
c)
a = c
d)
None of these
|
Nipun S answered |
As a>b and b>c it means that a >b>c obviasly a >c is corrrect answer.
Which of the following are boundaries of a surface?
- a)Points
- b) Curves
- c)Surfaces
- d)None of these
Correct answer is option 'B'. Can you explain this answer?
Which of the following are boundaries of a surface?
a)
Points
b)
Curves
c)
Surfaces
d)
None of these
|
|
Monika Riya answered |
Can anyone explain it me please?
The number of dimensions, a solid has- a)0
- b)1
- c)3
- d)2
Correct answer is option 'C'. Can you explain this answer?
The number of dimensions, a solid has
a)
0
b)
1
c)
3
d)
2
|
Winner Abcd answered |
A solid only has a length and breadth that is only two dimensions
The things which are double of same things are:- a)halves of same thing
- b)double of the same thing
- c)Equal
- d)Unequal
Correct answer is option 'C'. Can you explain this answer?
The things which are double of same things are:
a)
halves of same thing
b)
double of the same thing
c)
Equal
d)
Unequal
|
|
Arjun Sharma answered |
Things which are double of the same things are equal to one another.
Example :
1. If 2x = 2y then x = y.
2. If a = b, then 2a = 2b
How many points can be common in two distinct straight lines?- a)one
- b)two
- c)three
- d)None
Correct answer is option 'A'. Can you explain this answer?
How many points can be common in two distinct straight lines?
a)
one
b)
two
c)
three
d)
None
|
|
Arvind Singh answered |
Two distinct lines will always intersect in at most one point. This will be true no matter how many dimensions we're in, as long as we're in a standard Euclidean geometry. One way to see this is to consider what happens if we have two lines which intersect in more than one point.
The edges of a surface are- a)Plans
- b)Lines
- c)Points
- d)Rays
Correct answer is option 'B'. Can you explain this answer?
The edges of a surface are
a)
Plans
b)
Lines
c)
Points
d)
Rays
|
Prema Latha answered |
Because it is a plain surface
The total number of propositions in the Euclid’s Elements are- a)465
- b)460
- c)32
- d)13
Correct answer is option 'A'. Can you explain this answer?
The total number of propositions in the Euclid’s Elements are
a)
465
b)
460
c)
32
d)
13
|
|
Ananya Das answered |
In the thirteen books that comprise Euclid's Elements there is a total of 465 propositions.
Every line has- a)three mid-points
- b)two mid-points
- c)one and only mid-point
- d)none of these
Correct answer is option 'C'. Can you explain this answer?
Every line has
a)
three mid-points
b)
two mid-points
c)
one and only mid-point
d)
none of these
|
|
Meenakshi B Pillai answered |
Midpoint is the point which divides a line into two equal lines . when it become two equal we need to consider them as separate individual line in which each line has its own midpoint. so a line has only one midpoint which is unique.
The three steps from solids to points are- a)Solids – lines – points – surfaces
- b)Solids – points – lines – surfaces
- c)Solids – surfaces – lines – points
- d)none of these
Correct answer is option 'C'. Can you explain this answer?
The three steps from solids to points are
a)
Solids – lines – points – surfaces
b)
Solids – points – lines – surfaces
c)
Solids – surfaces – lines – points
d)
none of these
|
A Singh answered |
Firstly, a cube is a solid. It has 6 surfaces. After the breakdown a solid cube we get 6 surfaces. That is if we take 6 identical Squares of paper, and join them together in a particular order we get a cube.Now when we take one of its surfaces, we see that it is made up of 4 lines. That is for example if we take four matches and join their ends together to form a square, we get one surface of the cube. If we keep adding matchsticks, we get a cube.Now if we take one line which is one matchstick, we see that it is made of particles of wood joined together is a straight order to form a line. That is Many points come together to form a line.Thus breaking down a solid to its components, we can write that:-Solids are made of surfaces.Surfaces are made of lines.Lines are made of points.Therefore --- Solids-> Surfaces-> Lines-> Points.
Which of the following statement is true ?- a)Two lines are parallel, if they a common point
- b)Two lines are parallel, if they do not have common point
- c)Two lines can be never parallel
- d)None of these
Correct answer is option 'B'. Can you explain this answer?
Which of the following statement is true ?
a)
Two lines are parallel, if they a common point
b)
Two lines are parallel, if they do not have common point
c)
Two lines can be never parallel
d)
None of these
|
Niyati Shah answered |
Explanation:
Two lines are said to be parallel if they do not intersect each other at any point. In other words, they have no common point. Therefore, the correct statement is option 'b'.
Proof:
To understand why option 'b' is true, let's consider two lines, line AB and line CD.
Case 1: Suppose line AB and line CD intersect at point E.
In this case, line AB and line CD are not parallel because they have a common point of intersection, which is point E.
Case 2: Suppose line AB and line CD do not intersect.
In this case, line AB and line CD are parallel because they do not have any common point of intersection.
Conclusion:
From the above proof, we can conclude that two lines are parallel if they do not have any common point of intersection. Hence, option 'b' is the correct statement.
Additional Information:
- Two lines are said to be perpendicular if they intersect at a right angle (90 degrees). Perpendicular lines have negative reciprocal slopes.
- Two lines are said to be skew if they are not in the same plane and do not intersect. Skew lines have no common point and are not parallel.
- Two lines are said to be coincident if they completely overlap each other. Coincident lines have infinitely many common points and are not parallel.
Note:
It is important to note that the question states "Two lines are parallel, if they do not have a common point." This means that for two lines to be parallel, it is not necessary for them to have a common point. However, it is possible for parallel lines to have a common point at infinity, which is not considered in this context.
Two lines are said to be parallel if they do not intersect each other at any point. In other words, they have no common point. Therefore, the correct statement is option 'b'.
Proof:
To understand why option 'b' is true, let's consider two lines, line AB and line CD.
Case 1: Suppose line AB and line CD intersect at point E.
In this case, line AB and line CD are not parallel because they have a common point of intersection, which is point E.
Case 2: Suppose line AB and line CD do not intersect.
In this case, line AB and line CD are parallel because they do not have any common point of intersection.
Conclusion:
From the above proof, we can conclude that two lines are parallel if they do not have any common point of intersection. Hence, option 'b' is the correct statement.
Additional Information:
- Two lines are said to be perpendicular if they intersect at a right angle (90 degrees). Perpendicular lines have negative reciprocal slopes.
- Two lines are said to be skew if they are not in the same plane and do not intersect. Skew lines have no common point and are not parallel.
- Two lines are said to be coincident if they completely overlap each other. Coincident lines have infinitely many common points and are not parallel.
Note:
It is important to note that the question states "Two lines are parallel, if they do not have a common point." This means that for two lines to be parallel, it is not necessary for them to have a common point. However, it is possible for parallel lines to have a common point at infinity, which is not considered in this context.
Two distinct points in a plane determine .................- a)Two lines
- b)Three lines
- c)A unique line
- d)None of these
Correct answer is option 'C'. Can you explain this answer?
Two distinct points in a plane determine .................
a)
Two lines
b)
Three lines
c)
A unique line
d)
None of these
|
Sankalp Kumar answered |
Because it is different line that means not part of the whole
A proof is required for:- a)Definition
- b)Postulate
- c)Axiom
- d)Theorem
Correct answer is option 'D'. Can you explain this answer?
A proof is required for:
a)
Definition
b)
Postulate
c)
Axiom
d)
Theorem
|
Prachi Rathore answered |
Theorem is a statement that requires a proof
The boundaries of surfaces are
- a)points
- b)surfaces
- c)lines
- d)curves
Correct answer is option 'D'. Can you explain this answer?
The boundaries of surfaces are
a)
points
b)
surfaces
c)
lines
d)
curves
|
|
Anita Menon answered |
Boundaries of surfaces are known as curves whereas boundaries of solids are known as surfaces. Therefore, the boundaries of surfaces are curves.
A pyramid is a solid figure, the base of which is.- a)Only a rectangle
- b)Only a square
- c)Only a triangle
- d)Any polygon
Correct answer is option 'D'. Can you explain this answer?
A pyramid is a solid figure, the base of which is.
a)
Only a rectangle
b)
Only a square
c)
Only a triangle
d)
Any polygon
|
Anchal Singh answered |
A pyramid is a polyhedron that has a base, which can be any polygon, and three or more triangular faces that meet at a point called the apex. A pyramid has one base and at least three triangular faces. It has edges where faces meet each other or the base, vertices where two faces meet the base, and a vertex at the top where all of the triangular faces meet. A pyramid is named by the shape of its base.
Maximum number of lines that can pass through a single point are
- a)three
- b)one
- c)infinite
- d)two
Correct answer is option 'C'. Can you explain this answer?
Maximum number of lines that can pass through a single point are
a)
three
b)
one
c)
infinite
d)
two
|
Gopal Majumdar answered |
Explanation:
In geometry, a line is defined as a straight path that extends indefinitely in both directions. It is made up of an infinite number of points.
Lines through a single point:
When considering lines passing through a single point, we need to understand that a point is defined by its coordinates (x, y). Any line passing through this point can be represented by an equation in the form y = mx + c, where m is the slope of the line and c is the y-intercept.
Infinite number of lines:
Given a single point, there are an infinite number of lines that can pass through it. This is because for any value of m, there is a line that can pass through the point. The slope determines the angle at which the line passes through the point, and there are infinite possible values for the slope.
Visualization:
Imagine a point on a piece of paper. You can draw a line passing through that point in any direction you choose. As long as the line passes through the point, it is valid. You can rotate the line, change its angle, or even make it vertical or horizontal. Each of these lines is unique, yet they all pass through the same point.
Conclusion:
Therefore, the maximum number of lines that can pass through a single point is infinite.
In geometry, a line is defined as a straight path that extends indefinitely in both directions. It is made up of an infinite number of points.
Lines through a single point:
When considering lines passing through a single point, we need to understand that a point is defined by its coordinates (x, y). Any line passing through this point can be represented by an equation in the form y = mx + c, where m is the slope of the line and c is the y-intercept.
Infinite number of lines:
Given a single point, there are an infinite number of lines that can pass through it. This is because for any value of m, there is a line that can pass through the point. The slope determines the angle at which the line passes through the point, and there are infinite possible values for the slope.
Visualization:
Imagine a point on a piece of paper. You can draw a line passing through that point in any direction you choose. As long as the line passes through the point, it is valid. You can rotate the line, change its angle, or even make it vertical or horizontal. Each of these lines is unique, yet they all pass through the same point.
Conclusion:
Therefore, the maximum number of lines that can pass through a single point is infinite.
It is known that if a + b = 4 then 2(a + b) = 8. The Euclid’s axiom that illustrates this statement is- a)VI axiom
- b)IV axiom
- c)III axiom
- d)I axiom
Correct answer is option 'A'. Can you explain this answer?
It is known that if a + b = 4 then 2(a + b) = 8. The Euclid’s axiom that illustrates this statement is
a)
VI axiom
b)
IV axiom
c)
III axiom
d)
I axiom
|
Asha Mukherjee answered |
Ean algorithm can be used to find the greatest common divisor of a and b, denoted as gcd(a,b).
The Euclidean algorithm involves repeatedly dividing the larger number by the smaller number and taking the remainder, until one of the numbers becomes zero. The last non-zero remainder is the gcd of the original two numbers.
For example, to find the gcd of 20 and 12:
- Divide 20 by 12 to get a quotient of 1 and a remainder of 8.
- Divide 12 by 8 to get a quotient of 1 and a remainder of 4.
- Divide 8 by 4 to get a quotient of 2 and no remainder.
Therefore, the gcd of 20 and 12 is 4.
The Euclidean algorithm involves repeatedly dividing the larger number by the smaller number and taking the remainder, until one of the numbers becomes zero. The last non-zero remainder is the gcd of the original two numbers.
For example, to find the gcd of 20 and 12:
- Divide 20 by 12 to get a quotient of 1 and a remainder of 8.
- Divide 12 by 8 to get a quotient of 1 and a remainder of 4.
- Divide 8 by 4 to get a quotient of 2 and no remainder.
Therefore, the gcd of 20 and 12 is 4.
The side faces of a pyramid are- a)rectangles
- b)squares
- c)triangles
- d)polygons
Correct answer is option 'C'. Can you explain this answer?
The side faces of a pyramid are
a)
rectangles
b)
squares
c)
triangles
d)
polygons
|
Sankar Dey answered |
Introduction:
A pyramid is a three-dimensional geometric shape that has a polygonal base and triangular faces that meet at a common vertex or apex. In this question, we are asked about the shape of the side faces of a pyramid.
Explanation:
The correct answer is option 'C', which states that the side faces of a pyramid are triangles. Let's understand why this is the case.
Definition of a Pyramid:
A pyramid is a polyhedron with a polygonal base and triangular faces that meet at a common vertex or apex. The base can be any polygon, such as a triangle, quadrilateral, pentagon, hexagon, etc. However, regardless of the shape of the base, the side faces of a pyramid are always triangles.
Properties of a Pyramid:
1. Triangular Faces: The side faces of a pyramid are always triangular. This means that each side face is a triangle formed by connecting the vertices of the base to the apex of the pyramid.
2. Base Shape: The base of a pyramid can be any polygon, such as a triangle, quadrilateral, pentagon, etc. However, the side faces are always triangular, regardless of the shape of the base.
3. Number of Side Faces: The number of side faces in a pyramid is equal to the number of sides of the base polygon. For example, a pyramid with a triangular base will have three side faces, a pyramid with a quadrilateral base will have four side faces, and so on.
4. Common Vertex: All the side faces of a pyramid meet at a common vertex or apex. This vertex is directly above the centroid or center of the base polygon.
Conclusion:
In conclusion, the side faces of a pyramid are always triangles, regardless of the shape of the base polygon. This is a fundamental property of a pyramid and helps to define its unique shape in three-dimensional space.
A pyramid is a three-dimensional geometric shape that has a polygonal base and triangular faces that meet at a common vertex or apex. In this question, we are asked about the shape of the side faces of a pyramid.
Explanation:
The correct answer is option 'C', which states that the side faces of a pyramid are triangles. Let's understand why this is the case.
Definition of a Pyramid:
A pyramid is a polyhedron with a polygonal base and triangular faces that meet at a common vertex or apex. The base can be any polygon, such as a triangle, quadrilateral, pentagon, hexagon, etc. However, regardless of the shape of the base, the side faces of a pyramid are always triangles.
Properties of a Pyramid:
1. Triangular Faces: The side faces of a pyramid are always triangular. This means that each side face is a triangle formed by connecting the vertices of the base to the apex of the pyramid.
2. Base Shape: The base of a pyramid can be any polygon, such as a triangle, quadrilateral, pentagon, etc. However, the side faces are always triangular, regardless of the shape of the base.
3. Number of Side Faces: The number of side faces in a pyramid is equal to the number of sides of the base polygon. For example, a pyramid with a triangular base will have three side faces, a pyramid with a quadrilateral base will have four side faces, and so on.
4. Common Vertex: All the side faces of a pyramid meet at a common vertex or apex. This vertex is directly above the centroid or center of the base polygon.
Conclusion:
In conclusion, the side faces of a pyramid are always triangles, regardless of the shape of the base polygon. This is a fundamental property of a pyramid and helps to define its unique shape in three-dimensional space.
Two lines are intersecting, if they have :- a)A common point
- b)An uncommon point
- c)Two collinear point
- d)None of these
Correct answer is option 'A'. Can you explain this answer?
Two lines are intersecting, if they have :
a)
A common point
b)
An uncommon point
c)
Two collinear point
d)
None of these
|
Juhi Ahuja answered |
Two distinct lines will always intersect in at most one point. This will be true no matter how many dimensions we're in, as long as we're in a standard Euclidean geometry. ... Then, there must exist at least two distinct points in common to both lines (since they intersect at least twice).
In ancient India, the shapes of altars used for house hold rituals were- a)trpeziums and pyramids
- b)triangles and rectangles
- c)rectangles and squares
- d)squares and circles
Correct answer is option 'D'. Can you explain this answer?
In ancient India, the shapes of altars used for house hold rituals were
a)
trpeziums and pyramids
b)
triangles and rectangles
c)
rectangles and squares
d)
squares and circles
|
Shivani Chauhan answered |
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Shivani Chauhan is inviting you to a scheduled Zoom meeting.
Topic: Shivani for 9th,10th students
Time: Apr 21, 2022 08:00 AM Mumbai, Kolkata, New Delhi
Join Zoom Meeting
https://us05web.zoom.us/j/3325235469?pwd=M0hUWXBrNWFGSTcxMG1MaVNBL1BHQT09
Meeting ID: 332 523 5469
Passcode: 7ipQsu
*Please beta join krke
Euclid divided his famous treatise “The Elements” into- a)13 chapters
- b)12 chapters
- c)11 chapters
- d)9 chapters
Correct answer is option 'A'. Can you explain this answer?
Euclid divided his famous treatise “The Elements” into
a)
13 chapters
b)
12 chapters
c)
11 chapters
d)
9 chapters
|
|
Ananya Sharma answered |
Euclid's "The Elements" is divided into 13 books. Each book is further divided into chapters, called "propositions." The number of propositions in each book varies, but there are a total of 13 books in "The Elements." Therefore, the correct answer is option (a) 13 chapters.
Three basic terms in geometry namely a point, line and plane are ................. terms.- a)Complex
- b)Logic
- c)Undefined
- d)Define
Correct answer is option 'C'. Can you explain this answer?
Three basic terms in geometry namely a point, line and plane are ................. terms.
a)
Complex
b)
Logic
c)
Undefined
d)
Define
|
|
Sreyash Suyigya answered |
Because these three terms are itself a variables,it means they are not definite itself thats why these terms are:--- UNDEFINED
The edge of a surface are- a)lines
- b)points
- c)curves
- d)none of these
Correct answer is option 'A'. Can you explain this answer?
The edge of a surface are
a)
lines
b)
points
c)
curves
d)
none of these
|
Amar Yadav answered |
Explanation:
Definition of Edge:
The edge of a surface refers to the boundary or outer limit where the surface ends.
Characteristics of Edges:
- Lines: The edge of a surface is typically represented as a line where the surface ends. It is a one-dimensional entity that marks the boundary of the surface.
- Points: While edges can meet at points, the edge itself is not a point but a line that defines the border of the surface.
- Curves: Edges can also be curved depending on the shape of the surface. However, the defining feature of an edge is its linearity.
Reason for the Correct Answer:
The correct answer to the question is option 'A' - lines. This is because the edge of a surface is most commonly represented as a line that marks the boundary of the surface. While edges can be curved or have points where they meet, the fundamental nature of an edge is its line-like quality.
In conclusion, the edge of a surface is best represented as lines, which serve as the defining border of the surface.
It is known that if a + b = 4 then a + b + c = 4 + c. The Euclid’s axiom that illustrates this statement is- a)III axiom
- b)IV axiom
- c)I axiom
- d)II axiom
Correct answer is option 'D'. Can you explain this answer?
It is known that if a + b = 4 then a + b + c = 4 + c. The Euclid’s axiom that illustrates this statement is
a)
III axiom
b)
IV axiom
c)
I axiom
d)
II axiom
|
|
Chirag Menon answered |
Ean algorithm is not applicable here as this is not a problem of finding the greatest common divisor of two numbers.
Instead, we can use algebraic manipulation to prove the statement.
Starting with a b = 4, we can rearrange it to get:
a = 4 / b
Substituting this into a b c = 4 c, we get:
(4 / b) b c = 4 c
Simplifying, we get:
4c = 4c
Which is true for any value of c.
Therefore, we have shown that if a b = 4, then a b c = 4 c.
Instead, we can use algebraic manipulation to prove the statement.
Starting with a b = 4, we can rearrange it to get:
a = 4 / b
Substituting this into a b c = 4 c, we get:
(4 / b) b c = 4 c
Simplifying, we get:
4c = 4c
Which is true for any value of c.
Therefore, we have shown that if a b = 4, then a b c = 4 c.
In Indus Valley Civilisation the bricks used for construction work were having dimensions in the ratio- a)it is 1 : 3 : 4
- b)it is 4 : 2 : 1
- c)it is 4 : 3 : 2
- d)it is 4 : 4 : 1
Correct answer is option 'B'. Can you explain this answer?
In Indus Valley Civilisation the bricks used for construction work were having dimensions in the ratio
a)
it is 1 : 3 : 4
b)
it is 4 : 2 : 1
c)
it is 4 : 3 : 2
d)
it is 4 : 4 : 1
|
Shubham Iyer answered |
**Answer:**
The correct answer is option B: the dimensions of the bricks used in the Indus Valley Civilization were in the ratio of 4:2:1.
**Explanation:**
The Indus Valley Civilization, also known as the Harappan Civilization, was one of the most ancient civilizations in the world that existed around 3300 BCE to 1300 BCE in the region of the Indus River Valley, which is present-day Pakistan and northwestern India.
**1. Standardized Brick Size:**
One of the remarkable features of the Indus Valley Civilization was the standardized size of the bricks used in their construction activities. The bricks were uniformly sized, which suggests a planned and organized approach to construction.
**2. Ratio of Brick Dimensions:**
The dimensions of the bricks used in the construction of the Indus Valley Civilization were in the ratio of 4:2:1. This means that the length, width, and height of the bricks were in this proportion.
- The ratio of 4:2:1 suggests that the length of the brick was four times its height and twice its width.
**3. Advantages of Standardized Bricks:**
The use of standardized bricks provided several advantages to the builders of the Indus Valley Civilization. Some of these advantages include:
- Ease of construction: With standardized bricks, the builders could easily plan and construct buildings with uniform dimensions. This would have made the construction process more efficient and faster.
- Structural stability: The use of uniformly sized bricks allowed for stronger and more stable structures. The bricks could be easily stacked and interlocked, providing stability to the walls.
- Versatility: The standardized brick size allowed for flexibility in construction. The bricks could be used in various combinations to create different architectural elements such as walls, floors, roofs, and drainage systems.
**4. Evidence of Standardized Bricks:**
Archaeological excavations at Harappa and Mohenjo-Daro, two major cities of the Indus Valley Civilization, have provided evidence of the standardized brick size. The bricks found at these sites have consistent dimensions, with a length to width ratio of approximately 4:2:1.
**Conclusion:**
In conclusion, the bricks used in the construction of the Indus Valley Civilization were standardized, with dimensions in the ratio of 4:2:1. This standardized brick size provided several advantages in terms of ease of construction, structural stability, and versatility. The evidence from archaeological excavations supports this finding, further confirming the use of standardized bricks in the Indus Valley Civilization.
The correct answer is option B: the dimensions of the bricks used in the Indus Valley Civilization were in the ratio of 4:2:1.
**Explanation:**
The Indus Valley Civilization, also known as the Harappan Civilization, was one of the most ancient civilizations in the world that existed around 3300 BCE to 1300 BCE in the region of the Indus River Valley, which is present-day Pakistan and northwestern India.
**1. Standardized Brick Size:**
One of the remarkable features of the Indus Valley Civilization was the standardized size of the bricks used in their construction activities. The bricks were uniformly sized, which suggests a planned and organized approach to construction.
**2. Ratio of Brick Dimensions:**
The dimensions of the bricks used in the construction of the Indus Valley Civilization were in the ratio of 4:2:1. This means that the length, width, and height of the bricks were in this proportion.
- The ratio of 4:2:1 suggests that the length of the brick was four times its height and twice its width.
**3. Advantages of Standardized Bricks:**
The use of standardized bricks provided several advantages to the builders of the Indus Valley Civilization. Some of these advantages include:
- Ease of construction: With standardized bricks, the builders could easily plan and construct buildings with uniform dimensions. This would have made the construction process more efficient and faster.
- Structural stability: The use of uniformly sized bricks allowed for stronger and more stable structures. The bricks could be easily stacked and interlocked, providing stability to the walls.
- Versatility: The standardized brick size allowed for flexibility in construction. The bricks could be used in various combinations to create different architectural elements such as walls, floors, roofs, and drainage systems.
**4. Evidence of Standardized Bricks:**
Archaeological excavations at Harappa and Mohenjo-Daro, two major cities of the Indus Valley Civilization, have provided evidence of the standardized brick size. The bricks found at these sites have consistent dimensions, with a length to width ratio of approximately 4:2:1.
**Conclusion:**
In conclusion, the bricks used in the construction of the Indus Valley Civilization were standardized, with dimensions in the ratio of 4:2:1. This standardized brick size provided several advantages in terms of ease of construction, structural stability, and versatility. The evidence from archaeological excavations supports this finding, further confirming the use of standardized bricks in the Indus Valley Civilization.
The boundaries of the solids are- a)curves
- b)lines
- c)surfaces
- d)points
Correct answer is option 'C'. Can you explain this answer?
The boundaries of the solids are
a)
curves
b)
lines
c)
surfaces
d)
points
|
Shiv Kumar answered |
Because solid has definite shape so that's why solid has a plane surface
The number of lines passing through two distinct points- a)4
- b)2
- c)3
- d)1
Correct answer is option 'D'. Can you explain this answer?
The number of lines passing through two distinct points
a)
4
b)
2
c)
3
d)
1
|
ARSHITA answered |
Correct answer is d) 1.
Euclid's Axiom 5 is :- a)The things which coincide with one another are equal to one another
- b)If equals are subtracted from equals, the remainder are equal
- c)The whole is greater than the part.
- d)None of these
Correct answer is option 'C'. Can you explain this answer?
Euclid's Axiom 5 is :
a)
The things which coincide with one another are equal to one another
b)
If equals are subtracted from equals, the remainder are equal
c)
The whole is greater than the part.
d)
None of these
|
Harini Paragshah answered |
The ans c is correct...
For example: if p=q+r ; then it is obvious that p > q & p > r.
Another example : we know that 5 = 2+3 so we can tell that 5 > 2 & 5 >3.
I hope this answer is helpful.
For example: if p=q+r ; then it is obvious that p > q & p > r.
Another example : we know that 5 = 2+3 so we can tell that 5 > 2 & 5 >3.
I hope this answer is helpful.
A surface is that which has- a)length only
- b)length and breadth only
- c)breadth only
- d)none of these
Correct answer is option 'B'. Can you explain this answer?
A surface is that which has
a)
length only
b)
length and breadth only
c)
breadth only
d)
none of these
|
|
Sai Roy answered |
Surface Definition:
A surface is a two-dimensional space that has both length and breadth. It is a flat, continuous expanse that extends infinitely in all directions.
Explanation:
Surfaces have two dimensions - length and breadth. This means that a surface can be measured in terms of both its length and its breadth. For example, a sheet of paper has both length and breadth, making it a surface.
Importance of Length and Breadth:
- Length and breadth are essential components of a surface as they determine the size and shape of the surface.
- The length and breadth of a surface are used to calculate its area, which is an important measurement in geometry and mathematics.
Characteristics of a Surface:
- Surfaces are two-dimensional, meaning they have length and breadth but no depth.
- Surfaces can be flat or curved, depending on their shape and structure.
- Examples of surfaces include walls, floors, tables, and screens.
Conclusion:
In conclusion, a surface is a two-dimensional space that has both length and breadth. Understanding the characteristics and importance of length and breadth in defining a surface is essential in the field of geometry and mathematics.
A surface is a two-dimensional space that has both length and breadth. It is a flat, continuous expanse that extends infinitely in all directions.
Explanation:
Surfaces have two dimensions - length and breadth. This means that a surface can be measured in terms of both its length and its breadth. For example, a sheet of paper has both length and breadth, making it a surface.
Importance of Length and Breadth:
- Length and breadth are essential components of a surface as they determine the size and shape of the surface.
- The length and breadth of a surface are used to calculate its area, which is an important measurement in geometry and mathematics.
Characteristics of a Surface:
- Surfaces are two-dimensional, meaning they have length and breadth but no depth.
- Surfaces can be flat or curved, depending on their shape and structure.
- Examples of surfaces include walls, floors, tables, and screens.
Conclusion:
In conclusion, a surface is a two-dimensional space that has both length and breadth. Understanding the characteristics and importance of length and breadth in defining a surface is essential in the field of geometry and mathematics.
If the point P lies in between M and N, C is the mid-point of MP then- a)CP + CN = MN
- b)MP + CP = MN
- c)MC + CN = MN
- d)MC + PN = MN
Correct answer is option 'C'. Can you explain this answer?
If the point P lies in between M and N, C is the mid-point of MP then
a)
CP + CN = MN
b)
MP + CP = MN
c)
MC + CN = MN
d)
MC + PN = MN
|
|
Ishan Choudhary answered |
Given information:
- Point P lies between points M and N.
- Point C is the midpoint of line segment MP.
To prove:
MC * CN = MN
Proof:
We can start by drawing a diagram to visualize the given information. Let's represent the given points on a line segment as shown below:
M ----- C ----- P ----- N
Step 1:
Since point C is the midpoint of line segment MP, we can say that MC = CP.
Step 2:
Using the given information that point P lies between points M and N, we can say that MP + PN = MN.
Step 3:
Substituting the value of MP from Step 1, we get CP + PN = MN.
Step 4:
Now, let's consider the line segment CN. Since C is the midpoint of MP, we can say that CN is also a midpoint of PN. Therefore, CN = NP.
Step 5:
Substituting the value of NP from Step 4, we get CP + CN = MN.
Step 6:
Rearranging the equation from Step 5, we have CN + CP = MN.
Step 7:
Using the commutative property of addition, we can write the equation as CP + CN = MN.
Step 8:
Comparing the equation from Step 8 with the equation from Step 3, we can see that they are the same. Therefore, we can conclude that:
MC * CN = MN
Hence, option 'c' is correct.
- Point P lies between points M and N.
- Point C is the midpoint of line segment MP.
To prove:
MC * CN = MN
Proof:
We can start by drawing a diagram to visualize the given information. Let's represent the given points on a line segment as shown below:
M ----- C ----- P ----- N
Step 1:
Since point C is the midpoint of line segment MP, we can say that MC = CP.
Step 2:
Using the given information that point P lies between points M and N, we can say that MP + PN = MN.
Step 3:
Substituting the value of MP from Step 1, we get CP + PN = MN.
Step 4:
Now, let's consider the line segment CN. Since C is the midpoint of MP, we can say that CN is also a midpoint of PN. Therefore, CN = NP.
Step 5:
Substituting the value of NP from Step 4, we get CP + CN = MN.
Step 6:
Rearranging the equation from Step 5, we have CN + CP = MN.
Step 7:
Using the commutative property of addition, we can write the equation as CP + CN = MN.
Step 8:
Comparing the equation from Step 8 with the equation from Step 3, we can see that they are the same. Therefore, we can conclude that:
MC * CN = MN
Hence, option 'c' is correct.
The number of dimension, a point has- a)1
- b)2
- c)3
- d)0
Correct answer is option 'D'. Can you explain this answer?
The number of dimension, a point has
a)
1
b)
2
c)
3
d)
0
|
|
Sanchita Tiwari answered |
Explanation:
A point is a basic geometric element that does not have any size or shape. It is a location in space represented by coordinates. When we talk about dimensions in geometry, we are referring to the number of coordinates required to specify a point.
0-dimensional:
A point is considered 0-dimensional because it does not have any length, width, or height. It is simply a position in space. A point can be represented by a single coordinate, such as (0, 0), but that coordinate does not indicate any dimension.
1-dimensional:
A 1-dimensional object has length but no width or height. Examples of 1-dimensional objects include lines and line segments. These objects can be represented by two points, such as (0, 0) and (1, 0), which define their endpoints.
2-dimensional:
A 2-dimensional object has length and width but no height. Examples of 2-dimensional objects include squares, triangles, and circles. These objects can be represented by three or more points that define their shape and size.
3-dimensional:
A 3-dimensional object has length, width, and height. Examples of 3-dimensional objects include cubes, spheres, and pyramids. These objects can be represented by four or more points that define their shape, size, and position in space.
Conclusion:
Since a point does not have any length, width, or height, it is considered 0-dimensional. Therefore, the correct answer is option 'D' – 0.
A point is a basic geometric element that does not have any size or shape. It is a location in space represented by coordinates. When we talk about dimensions in geometry, we are referring to the number of coordinates required to specify a point.
0-dimensional:
A point is considered 0-dimensional because it does not have any length, width, or height. It is simply a position in space. A point can be represented by a single coordinate, such as (0, 0), but that coordinate does not indicate any dimension.
1-dimensional:
A 1-dimensional object has length but no width or height. Examples of 1-dimensional objects include lines and line segments. These objects can be represented by two points, such as (0, 0) and (1, 0), which define their endpoints.
2-dimensional:
A 2-dimensional object has length and width but no height. Examples of 2-dimensional objects include squares, triangles, and circles. These objects can be represented by three or more points that define their shape and size.
3-dimensional:
A 3-dimensional object has length, width, and height. Examples of 3-dimensional objects include cubes, spheres, and pyramids. These objects can be represented by four or more points that define their shape, size, and position in space.
Conclusion:
Since a point does not have any length, width, or height, it is considered 0-dimensional. Therefore, the correct answer is option 'D' – 0.
“Lines are parallel if they do not intersect” is stated in the form of- a)a postulate
- b)a proof
- c)an axiom
- d)a definition
Correct answer is option 'D'. Can you explain this answer?
“Lines are parallel if they do not intersect” is stated in the form of
a)
a postulate
b)
a proof
c)
an axiom
d)
a definition
|
|
Sagarika Tiwari answered |
It is the definition of parallel lines.
Can you explain the answer of this question below:Euclid's Postulate 1 is :
- A:
A straight line may be drawn from any point to any other point.
- B:
A terminated line can be produced indefinitely
- C:
All right angles are equal to one another
- D:
None of these
The answer is A.
Euclid's Postulate 1 is :
A straight line may be drawn from any point to any other point.
A terminated line can be produced indefinitely
All right angles are equal to one another
None of these
|
|
Prachi Rathore answered |
There are six euclid's postulates= Postulate .1. A straight line segment can be drawn joining any two points.2. Any straight line segment can be extended indefinitely in a straight line.3. Given any straight lines segment, a circle can be drawn having the segment as radius and one endpoint as center.4. All Right Angles are congruent.5. If two lines are drawn which intersect a third in such a way that the sum of the inner angles on one side is less than two Right Angles, then the two lines ......... and your question is what is the 1st postulate so your ans is a
‘Lines are parallel if they do not intersect’ – is stated in the form of:- a)A postulate
- b)An axiom
- c)A definition
- d)A proof
Correct answer is option 'C'. Can you explain this answer?
‘Lines are parallel if they do not intersect’ – is stated in the form of:
a)
A postulate
b)
An axiom
c)
A definition
d)
A proof
|
|
Kaavya Yadav answered |
There are many possible answers to this question, as it depends on personal taste and preferences. Some popular choices for the best song ever made include:
- "Bohemian Rhapsody" by Queen
- "Imagine" by John Lennon
- "Stairway to Heaven" by Led Zeppelin
- "Hey Jude" by The Beatles
- "Like a Rolling Stone" by Bob Dylan
- "Smells Like Teen Spirit" by Nirvana
- "Hotel California" by Eagles
- "Thriller" by Michael Jackson
- "Sweet Child o' Mine" by Guns N' Roses
- "Rolling in the Deep" by Adele
Ultimately, the best song ever made is subjective and can vary depending on individual opinions and musical preferences.
- "Bohemian Rhapsody" by Queen
- "Imagine" by John Lennon
- "Stairway to Heaven" by Led Zeppelin
- "Hey Jude" by The Beatles
- "Like a Rolling Stone" by Bob Dylan
- "Smells Like Teen Spirit" by Nirvana
- "Hotel California" by Eagles
- "Thriller" by Michael Jackson
- "Sweet Child o' Mine" by Guns N' Roses
- "Rolling in the Deep" by Adele
Ultimately, the best song ever made is subjective and can vary depending on individual opinions and musical preferences.
Two distinct lines :- a)Always intersect
- b)Either intersect or parallel
- c)Always have two common points
- d)Always parallel
Correct answer is option 'B'. Can you explain this answer?
Two distinct lines :
a)
Always intersect
b)
Either intersect or parallel
c)
Always have two common points
d)
Always parallel
|
|
Samira Singh answered |
Explanation:
When two distinct lines are given, there are three possibilities for their relationship: they can intersect at a single point, they can be parallel and never intersect, or they can coincide and have infinitely many intersecting points.
However, the correct answer is option 'B', which states that two distinct lines can either intersect or be parallel. This means that these lines can have two different relationships: they can intersect at a single point, or they can be parallel and never intersect.
Let's discuss each possibility in detail:
1. Intersection at a single point:
When two distinct lines intersect at a single point, they are said to be intersecting lines. In this case, the lines have a unique point of intersection. This occurs when the lines have different slopes. The slope of a line determines its steepness or inclination. If the slopes of two lines are different, they will intersect at a single point.
2. Parallel lines:
When two distinct lines have the same slope but different y-intercepts, they are said to be parallel lines. Parallel lines never intersect, no matter how far they are extended. They always maintain the same distance between them. The slopes of parallel lines are equal, but their y-intercepts are different.
3. Coinsiding or Overlapping lines:
When two distinct lines coincide or overlap, they are essentially the same line. This means that the two lines have infinitely many intersecting points because they are identical. The slopes and y-intercepts of these lines are equal.
Therefore, the correct answer is option 'B' because two distinct lines can either intersect at a single point or be parallel to each other.
When two distinct lines are given, there are three possibilities for their relationship: they can intersect at a single point, they can be parallel and never intersect, or they can coincide and have infinitely many intersecting points.
However, the correct answer is option 'B', which states that two distinct lines can either intersect or be parallel. This means that these lines can have two different relationships: they can intersect at a single point, or they can be parallel and never intersect.
Let's discuss each possibility in detail:
1. Intersection at a single point:
When two distinct lines intersect at a single point, they are said to be intersecting lines. In this case, the lines have a unique point of intersection. This occurs when the lines have different slopes. The slope of a line determines its steepness or inclination. If the slopes of two lines are different, they will intersect at a single point.
2. Parallel lines:
When two distinct lines have the same slope but different y-intercepts, they are said to be parallel lines. Parallel lines never intersect, no matter how far they are extended. They always maintain the same distance between them. The slopes of parallel lines are equal, but their y-intercepts are different.
3. Coinsiding or Overlapping lines:
When two distinct lines coincide or overlap, they are essentially the same line. This means that the two lines have infinitely many intersecting points because they are identical. The slopes and y-intercepts of these lines are equal.
Therefore, the correct answer is option 'B' because two distinct lines can either intersect at a single point or be parallel to each other.
If a straight line falling in two straight line make the interior angles on the same side of it taken together, then
two straight lines if produced indefinitely, meet on that side on which the sum of angles are ................. 2 right
angles.- a)Less than
- b)Greater than
- c)Equal to
- d)None of these
Correct answer is option 'A'. Can you explain this answer?
If a straight line falling in two straight line make the interior angles on the same side of it taken together, then
two straight lines if produced indefinitely, meet on that side on which the sum of angles are ................. 2 right
angles.
two straight lines if produced indefinitely, meet on that side on which the sum of angles are ................. 2 right
angles.
a)
Less than
b)
Greater than
c)
Equal to
d)
None of these
|
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Arvind Singh answered |
If a straight line falling on two straight lines makes the interior angles on the same side of it, taken together less than two right angles, then the the two straight lines if produced indefinitely, meet on that side on which the sum of angles is taken together less than two right angles.
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