## Geometry Chapter 2 Points Lines Line Segment Ray And Their Concepts

**Geometry Chapter 2 Ponts**

- If a piece of paper is folded twice in two ways, then we find, there are two long straight markings along the creases, and the place or the position where they intersect is called a
**Point.** - Again the intersection of two adjacent edges of a page of a book or the upper surface of a square or rectangular table produces a point.
- In the above examples, we can take two long straight markings and two adjacent edges of a page or the upper surface of a table as two straight lines (which we shall discuss in the next article) and they intersect at a point.
- We know that a line has only length and its dimension is one.
- If we gradually diminish the length of a line, then in the extreme case, the length of the line can not
- be measured and the line is reduced to a point.
- As the length of a point is not measurable, the point has no dimension.
- So we can give the following definition of a point:
- That, which has only position but no dimension is called a point.
- A point has no length, breadth, or thickness.
- Although a geometrical point can not be drawn, a point can be drawn by pressing lightly the sharp end of a pencil on a piece of paper.
- A dot mark (.) which is produced on the paper is the required point.
- In geometry, in order to distinguish the different points from one another, we denote them by the capital letters of the English Alphabet such as A, B, D, D, E, F, etc.

**In this connection, the followings are to be kept in mind:**

- Two straight lines (which are not parallel) intersect at a point.
- A line can be regarded as a continuous series of innumerable points.
- By joining any two points, we can draw a line segment.
If a straight line is obtained by joining consecutively three or more points lying on the same plane i.e., if three or more points be situated on a straight line, then the points are called collinear points.**Collinear points:**- For example:
- Here the points A, B, C, and D lie on the straight line AD===; so the points A, B, C, and D are collinear points.

## Geometry Chapter 2 Lines

- The geometrical figure which has only length, but no breadth or width is called a line:
- For example, in the figure below both AB=== and PQ=== lines.
- Lines are classified into two types, (i) Straight lines, and (ii) Curved lines.

**Straight lines and curved lines :**

- In the previous chapter, we discussed lines only.
- A line is a geometrical figure which has only length but no breadth or thickness.
- It is a one-dimensional figure.
**There are two types of lines:**straight lines and curved lines.

**Now we shall discuss some examples:**

- If we stretch both ends of a thread till it becomes straight, then the figure that the stretched thread forms are a straight line.
- Two walls of a room meet at a line and also each of the walls of a room meets with the floor of the room at a line.
- These lines are called straight lines.
- If you walk along a circular track, then you will continuously change the direction of your movement.
- Suppose you start your walking towards the south, after some time you will be moving towards the east and afterward you may stand facing north, etc.
- The circular track along which you are walking is a curved line.
- But if you walk along a straight line facing say, south, then all along you would have faced south.
- Suppose, there are two stations located at A and B. There are innumerable ways of going from station A to station B as shown in the figure (among them, one is a straight line or way).
- If you want to go from A to B through the shortest possible route, then obviously you will have to go through the straight line route and this is only the straight route.
- All other routes are curved.
- From this, we conclude that through two points only one
straight line can be drawn and innumerable curved lines can be drawn.**curved route** A-line, whose one end can be reached from the other end without changing direction, is called a straight line.**Definition:**- A straight line can also be defined in the following way:
- A straight line is a line that can be extended on both sides uniformly without changing direction.
- In the adjacent figure, AB is a straight line.
- Definition: A line that is not a straight line or a line, whose one end can be reached from the other end by changing direction, is called a curved line.
- A curved line can also be defined in the following way:
- A curved line is a line that gradually deviated from
- A B is straight for some or all of its length.
- A curved line has many directions.
- Curved lines if the sharp ends of a pencil along the side of a scale placed on the surface of a paper, a straight line is obtained.
- The edges of a page, benches, tables, etc. are examples of straight lines.
- If we draw the sharp end of a pencil along the side of a coin placed on the surface of the paper, a curved line is obtained.
- The lines drawn on the surface of a sphere, cone, or cylinder are curved lines.

**Properties of straight lines:**

- Innumerable straight lines can be drawn through a point.
- Let O be a point on the plane of the paper.
- A, B, C, D, E, F, G be any number of points
- on the plane of the paper at which point O lies.
- We join these points with O, we get innumerable
- straight lines OA, OB, OC, OD, OE, OF, OG…….
- Thus through any point, we can draw as many straight lines as we, please.
- One and only one straight line can be drawn through two given points.
- Let A and B be two given points on a paper.
- Then only one straight line AB can be drawn through A and B.
- There is an infinite number of points on a line.
- An infinite number of points lie on line AB.
- The points which line on a line are called
**collinear** - Here P, A, B, C, D, and Q are collinear points because they lie on the same line.
- Three or more points may or may not lie on a line.
- Here the points A, B, C, D, and E lie on a line.
- Here the points P, Q, and R do not lie on the same line.
- If two straight lines intersect, then they must intersect at a point only.
- Here AB and CD be two straight lines and they intersect at a point P only.
- A curved line and a straight line intersect at more than one point.
- Here the curved line PQ intersects the straight line AB at four points X, Y, Z, and T.
- A straight line is not always drawn through any three given points.
- The maximum number of straight lines that can be drawn through three non-collinear points is three.
- Here A, B, and C are given three non-collinear points.
- Through them, only 3 lines AB, BC, and CA can be drawn.
- Two straight lines lying on the same plane may or may not intersect each other.
- If they intersect, then their point of intersection is only one.
- If they do not intersect then the straight lines are parallel.
- The opposite edges of a table, book, and brick are parallel to each other. even when they are extended to infinity on both sides, then they are said to be parallel straight lines.

**Two Or More Straight lines may or may not lie on the same plane:**

- Here AB, CD, and EF are three straight lines and they do not intersect when they are extended in both ways. So AB, CD, and EF are parallel to each other.
- If three or more straight lines lying on the same plane intersect at a point i.e. three or more straight lines pass through a single point, then they are said to be concurrent straight lines.
- Here AB, CD, EF, and GH pass through the same point O.
- Hence these lines are concurrent.
- Point O is called the point of concurrence.
- If the given straight lines do not pass through a single point i.e., if they do not meet at a point, then the straight lines are said to be not concurrent.
- Here AB, CD, EF do not meet at a point and so they are not concurrent.
- Two or more straight lines may or may not lie on the same plane.
- If two or more straight lines lie on the same plane they are said to be
**coplanar lines.** - If two or more straight lines do not lie on the same plane, then they are said to be
**non-coplanar lines.** - Two straight lines which do not lie on the same plane and they neither intersect nor parallel are called skew lines.

## Geometry Chapter 2 Line Segments

- A line segment is a bounded segment or a portion of a straight line by two fixed points.
- These two fixed points are called the endpoints of the line segment.
- Let A and B be two points on the straight line AB as shown in the figure below.
- The straight line AB can be extended on two sides (on the left and right sides) but AB is a bounded portion or segment of the straight line.
- This portion is bounded by points A and B.
- So, AB is a line segment. These two points A and B are the endpoints of the line segment AB.
- As line segment AB is bounded by the two fixed points A and B, the length of line segment AB can be measured.
- We generally denote the straight line AB by AB and the line segment AB by AB.
- The arrowheads are placed at the two ends of the straight line AB in order to mean that the straight line can be extended on both sides indefinitely.
- A and B are not the actual endpoints of the straight line.
- But in general, it is understood from the context, by AB we mean the line segment AB.
- We name the line segment according to the name of their endpoints.
- Let A, B, C, and D be four fixed points on the same straight line as shown in the figure below.
- The line segments are AB, BC, CD, AC, BD, and AD.
- Each edge of the surface of a table, almirah, length, and breadth of a room, each side of a rectangle, square, parallelogram, each side of a book, etc. are examples of line segments

## Geometry Chapter 2 Ray

- In a straight line, both ends of a line segment are extended indefinitely.
- Keeping one end of a line segment fixed, the other end is extended indefinitely, then it is called a Ray.
- In (1), AB is a line segment; in (2) the line segment AB is extended on both sides and so it is a line.
- In (3) A is fixed and the other end B is extended indefinitely.
- It is a ray.
- Again, in the end, B is fixed and the other end A is extended indefinitely.
- It is also a ray.
- In(3) and (4), arrowheads are given on the right-hand and left-hand sides only.
- The rays are represented by X by AB and BA respectively.
- The fixed end of a ray is called its vertex.
- For the ray OX, O is its vertex and the end X is extended indefinitely.

## Geometry Chapter 2 Distinguish Among Straight Lines Line Segments Ray

## Geometry Chapter 2 Properties Regarding Points Segments And Rays

- An infinite number of straight lines can be drawn through a fixed point.
- One and only one straight line can be drawn through two given fixed points.
- An infinite number of curved lines can be drawn through two given fixed points.
- There are an infinite number of points on a straight line or a curved line.
- Two straight lines lying on the same plane either are parallel or intersect at a point.
- Three or more points may lie on a straight line or may not lie on a straight line.

If the points lie on a line then they are said to be**Collinear Points.** - The maximum number of straight lines that can be drawn through, three non-collinear points is three.
- An Infinite number of rays can be drawn through a given point.
- we extinct then indefinitely on both sides, then they are said to be parallel to each other and the straight lines are said to be parallel straight lines.

**Concurrent straight lines:**

If three or more straight lines lying on the same plane intersect at a point i.e., three or more straight lines pass through a single point then they are said to be concurrent straight lines.