Introduction to Euclid's Geometry RD Sharma Solutions | Mathematics (Maths) Class 9 PDF Download

 Page 1


Question:1
Define the following terms:
(i) Line segment
(ii) Collinear points
(iii) Parallel lines
(iv) Intersecting lines
(v) Concurrent lines
(vi) Ray
(vii) Half-line.
Solution:
The given problem requires definitions of various terms.
(i) Line segment: 
A line segment AB can be defined as the part of the line with end points A and B, whereA and B are the two points
of the line.
It is denoted by 
Let us take a line with two points A and B
 
 
This is a line AB
 
While,
This is a line segment AB.
 
(ii) Collinear points:
When three or more points lie on the same line; they are said to be collinear.
Example: 
Let us take a line l. P, Q, R points lie on it.
 
So,
Page 2


Question:1
Define the following terms:
(i) Line segment
(ii) Collinear points
(iii) Parallel lines
(iv) Intersecting lines
(v) Concurrent lines
(vi) Ray
(vii) Half-line.
Solution:
The given problem requires definitions of various terms.
(i) Line segment: 
A line segment AB can be defined as the part of the line with end points A and B, whereA and B are the two points
of the line.
It is denoted by 
Let us take a line with two points A and B
 
 
This is a line AB
 
While,
This is a line segment AB.
 
(ii) Collinear points:
When three or more points lie on the same line; they are said to be collinear.
Example: 
Let us take a line l. P, Q, R points lie on it.
 
So,
Here, P, Q and R are collinear points.
 
(iii) Parallel lines:
Two or more lines are said to be parallel to each other if there is no point of intersection between them.
For Example:
Since, there is no point of intersection between l and m, they are parallel.
 
(iv) Intersecting lines:
Two or more lines are said to be intersecting lines if they meet each other at a point or they have a common point.
For Example:
 
l and m are the two lines both passing through point O. Hence, they are intersecting lines.
 
(v) Concurrent lines:
Two or more lines are said to be concurrent if they all pass through a common point or there exist a point common
to all of them.
 
For Example:
 
m, n, o and p are concurrent as they all have a common point O.
 
(vi) Ray:
A ray is defined as the part of the line with one end point such that it can be extended infinitely in the other
direction.
Page 3


Question:1
Define the following terms:
(i) Line segment
(ii) Collinear points
(iii) Parallel lines
(iv) Intersecting lines
(v) Concurrent lines
(vi) Ray
(vii) Half-line.
Solution:
The given problem requires definitions of various terms.
(i) Line segment: 
A line segment AB can be defined as the part of the line with end points A and B, whereA and B are the two points
of the line.
It is denoted by 
Let us take a line with two points A and B
 
 
This is a line AB
 
While,
This is a line segment AB.
 
(ii) Collinear points:
When three or more points lie on the same line; they are said to be collinear.
Example: 
Let us take a line l. P, Q, R points lie on it.
 
So,
Here, P, Q and R are collinear points.
 
(iii) Parallel lines:
Two or more lines are said to be parallel to each other if there is no point of intersection between them.
For Example:
Since, there is no point of intersection between l and m, they are parallel.
 
(iv) Intersecting lines:
Two or more lines are said to be intersecting lines if they meet each other at a point or they have a common point.
For Example:
 
l and m are the two lines both passing through point O. Hence, they are intersecting lines.
 
(v) Concurrent lines:
Two or more lines are said to be concurrent if they all pass through a common point or there exist a point common
to all of them.
 
For Example:
 
m, n, o and p are concurrent as they all have a common point O.
 
(vi) Ray:
A ray is defined as the part of the line with one end point such that it can be extended infinitely in the other
direction.
It is represented by 
For Example:
Here,  is a ray as it has one end point A and it can be extended indefinitely in other direction.
 
(vii) Half-line: 
A half-line can be defined as a part of the line which has one end point and extends indefinitely in the other
direction. It is different from ray as the end point is not included in the half-line.
 
For example,
When A is included in the part, then it is called a ray AB, but when A is not included then is called a half-line AB.
Question:2
(i) How many lines can pass through a given point?
(ii) In how many points can two distinct lines at the most intersect?
Solution:
(i) Let us take a point A.
If we try to draw lines passing through this point A, we can see that we can draw many lines. 
 
Therefore, infinite number of lines can pass through a given point.
 
(ii) Let us take two lines l and m, and intersect them.
 
Page 4


Question:1
Define the following terms:
(i) Line segment
(ii) Collinear points
(iii) Parallel lines
(iv) Intersecting lines
(v) Concurrent lines
(vi) Ray
(vii) Half-line.
Solution:
The given problem requires definitions of various terms.
(i) Line segment: 
A line segment AB can be defined as the part of the line with end points A and B, whereA and B are the two points
of the line.
It is denoted by 
Let us take a line with two points A and B
 
 
This is a line AB
 
While,
This is a line segment AB.
 
(ii) Collinear points:
When three or more points lie on the same line; they are said to be collinear.
Example: 
Let us take a line l. P, Q, R points lie on it.
 
So,
Here, P, Q and R are collinear points.
 
(iii) Parallel lines:
Two or more lines are said to be parallel to each other if there is no point of intersection between them.
For Example:
Since, there is no point of intersection between l and m, they are parallel.
 
(iv) Intersecting lines:
Two or more lines are said to be intersecting lines if they meet each other at a point or they have a common point.
For Example:
 
l and m are the two lines both passing through point O. Hence, they are intersecting lines.
 
(v) Concurrent lines:
Two or more lines are said to be concurrent if they all pass through a common point or there exist a point common
to all of them.
 
For Example:
 
m, n, o and p are concurrent as they all have a common point O.
 
(vi) Ray:
A ray is defined as the part of the line with one end point such that it can be extended infinitely in the other
direction.
It is represented by 
For Example:
Here,  is a ray as it has one end point A and it can be extended indefinitely in other direction.
 
(vii) Half-line: 
A half-line can be defined as a part of the line which has one end point and extends indefinitely in the other
direction. It is different from ray as the end point is not included in the half-line.
 
For example,
When A is included in the part, then it is called a ray AB, but when A is not included then is called a half-line AB.
Question:2
(i) How many lines can pass through a given point?
(ii) In how many points can two distinct lines at the most intersect?
Solution:
(i) Let us take a point A.
If we try to draw lines passing through this point A, we can see that we can draw many lines. 
 
Therefore, infinite number of lines can pass through a given point.
 
(ii) Let us take two lines l and m, and intersect them.
 
As, we can see here the two lines have only one point in common that is O.
Therefore, there is only one point where two distinct lines can intersect.
Question:3
(i) Given two points P and Q, find how many line segments do they deter-mine.
(ii) Name the line segments determined by the three collinear points P, Q and R.
Solution:
(i) In this problem we are given two points P and Q.
 
If we try to join these two points through a line segment, we can see that there can be only one such line segment
PQ.
 
Therefore, given two points, only one line segment is determined by them.
 
 
(ii) In the given problem, we are given three collinear points P, Q and R. Collinear points lie on the same line, so
they can be represented as
 
 
So, the various line segments determined here are P Q, Q R and P R.
Question:4
Write the turth value (T/F) of each of the following statements:
(i) Two lines intersect in a point.
(ii) Two lines may intersect in two points.
(iii) A segment has no length.
(iv) Two distinct points always determine a line.
(v) Every ray has a finite length.
(vi) A ray has one end-point only.
(vii) A segment has one end-point only.
(viii) The ray AB is same as ray BA.
(ix) Only a single line may pass through a given point.
Page 5


Question:1
Define the following terms:
(i) Line segment
(ii) Collinear points
(iii) Parallel lines
(iv) Intersecting lines
(v) Concurrent lines
(vi) Ray
(vii) Half-line.
Solution:
The given problem requires definitions of various terms.
(i) Line segment: 
A line segment AB can be defined as the part of the line with end points A and B, whereA and B are the two points
of the line.
It is denoted by 
Let us take a line with two points A and B
 
 
This is a line AB
 
While,
This is a line segment AB.
 
(ii) Collinear points:
When three or more points lie on the same line; they are said to be collinear.
Example: 
Let us take a line l. P, Q, R points lie on it.
 
So,
Here, P, Q and R are collinear points.
 
(iii) Parallel lines:
Two or more lines are said to be parallel to each other if there is no point of intersection between them.
For Example:
Since, there is no point of intersection between l and m, they are parallel.
 
(iv) Intersecting lines:
Two or more lines are said to be intersecting lines if they meet each other at a point or they have a common point.
For Example:
 
l and m are the two lines both passing through point O. Hence, they are intersecting lines.
 
(v) Concurrent lines:
Two or more lines are said to be concurrent if they all pass through a common point or there exist a point common
to all of them.
 
For Example:
 
m, n, o and p are concurrent as they all have a common point O.
 
(vi) Ray:
A ray is defined as the part of the line with one end point such that it can be extended infinitely in the other
direction.
It is represented by 
For Example:
Here,  is a ray as it has one end point A and it can be extended indefinitely in other direction.
 
(vii) Half-line: 
A half-line can be defined as a part of the line which has one end point and extends indefinitely in the other
direction. It is different from ray as the end point is not included in the half-line.
 
For example,
When A is included in the part, then it is called a ray AB, but when A is not included then is called a half-line AB.
Question:2
(i) How many lines can pass through a given point?
(ii) In how many points can two distinct lines at the most intersect?
Solution:
(i) Let us take a point A.
If we try to draw lines passing through this point A, we can see that we can draw many lines. 
 
Therefore, infinite number of lines can pass through a given point.
 
(ii) Let us take two lines l and m, and intersect them.
 
As, we can see here the two lines have only one point in common that is O.
Therefore, there is only one point where two distinct lines can intersect.
Question:3
(i) Given two points P and Q, find how many line segments do they deter-mine.
(ii) Name the line segments determined by the three collinear points P, Q and R.
Solution:
(i) In this problem we are given two points P and Q.
 
If we try to join these two points through a line segment, we can see that there can be only one such line segment
PQ.
 
Therefore, given two points, only one line segment is determined by them.
 
 
(ii) In the given problem, we are given three collinear points P, Q and R. Collinear points lie on the same line, so
they can be represented as
 
 
So, the various line segments determined here are P Q, Q R and P R.
Question:4
Write the turth value (T/F) of each of the following statements:
(i) Two lines intersect in a point.
(ii) Two lines may intersect in two points.
(iii) A segment has no length.
(iv) Two distinct points always determine a line.
(v) Every ray has a finite length.
(vi) A ray has one end-point only.
(vii) A segment has one end-point only.
(viii) The ray AB is same as ray BA.
(ix) Only a single line may pass through a given point.
(x) Two lines are coincident if they have only one point in common.
Solution:
(i) False
The given statement is false, as it is not necessary that the two lines always intersect. In the case of parallel lines,
the two lines never intersect or there is no common point between them.
 
(ii) False
The given statement is false, as there is only one point common between two intersecting lines. So, two lines
cannot intersect at two points.
 
(iii) False
A line segment is a part of line defined by two end points, so it is always of a definite length. Hence, the given
statement is false.
 
(iv) True
The given statement is true as a unique line can be determined by minimum of two distinct points.
 
(v) False
A ray is defined as a part of the line with one end point, where the other end can be stretched indefinitely. Therefore,
a ray cannot have a finite length.
 
(vi) True
The given statement is true as a ray is defined as a part of the line with one end point, where the other end can be
stretched indefinitely.
 
(vii) False
A line segment is a part of line defined by two end points. So the given statement is false as a line segment has
two end points.
 
(viii) False
While denoting a ray, the first letter denotes the end point and the second one denotes the end which can be
extended. So, for ray AB, A is the end point and B is the end which can be stretched while for ray BA, B is the end
point and A the end which can be stretched. Therefore, ray AB is not the same as ray BA.
 
(ix) False
The given statement is false as infinite number of lines can pass through a given point.
 
(x) False
Two lines are said to be coincident if they lie exactly on top of each other, which means that they have all the