The Basic Elements of Geometry
You are familiar with some terms like triangle, square, cube, cuboid, etc. These are examples of geometrical figures. To make these geometrical figures we need to know some basic elements. These basic elements are as follows:
Point(.): A point gives an idea of a location, by making a dot by a sharp pencil on a paper. It has no length, breadth, or thickness. It has just a position and only its location can be determined.
A point is denoted by a capital letter of the alphabet like A, B, C, etc.
For example: . P (This is point P.)
Line (↔): A line is a collection of points, which can be extended endlessly on both the sides. It has length only. It has neither breadth nor thickness.
A line can be denoted in two ways
(i) Denote it by a small letter of the alphabet like line l as shown in Fig.
(ii) Mark two points (say A and B) on the line as shown in Fig. and denote it by (overleftrightarrow{text{AB}}) or line AB.
Line Segment ( ¯ ): A Line segment is a part of a line that is bounded by two distinct end points. It has a definite length but no breadth and thickness. It is the shortest distance of any two points.
A line segment from A to B is represented by seg AB or (overline{text{AB}}) or (overline{text{BA}}).
This is seg AB or (overline{text{AB}}).
Ray (→): A ray is also a part of a line which has only one end point and can be extended endlessly in one direction. A ray has no breadth or thickness.
A ray is represented by (overrightarrow{AB}). It shows that A is the fixed point and B is a point on the path of a ray.
Light coming from the sun or torch is an example of a ray.
Comparison between line, line segment, and ray
Table shows the comparison between line, line segment, and ray.
Line |
Line Segment |
Ray |
1. A line has no definite length, | A line segment has definite length. | A ray has no length. |
2. A line has no end points. | A line segment has two end points. | A ray has one end point. |
3. A line has no thickness. | A line segment has no thickness. | A ray has no thickness. |
4. A line AB is represented by (overleftrightarrow{text{AB}}) | A line segment AB is represented by (overline{text{AB}}). | A ray AB is represented by (overrightarrow{AB}). |
Plane: A plane is a flat smooth surface that extends indefinitely in all directions. It has length and breadth but no thickness.
The top of a table, top and bottom of a cylinder, surface of a blackboard, etc. give the idea of a plane.
A plane can be denoted by taking three or more points on it, which do not lie on the same line.
Incidence properties in a plane
The relationship between a point and a line in a plane is called the incidence property. It states that
- Infinite number of lines can be drawn passing through a given fixed point in a plane.
Lines l_{1}, l_{2}, l_{3}, l_{4}, …… all pass through a point A. - One and only one line can be drawn passing through two given points in a plane.
If A and B are the two points in a plane, then l_{2 }becomes the unique line that passes through the points A and B.
COLLINEAR POINTS
Three or more points are said to be collinear, if they lie on the same line in a plane. This line is called the line of collinearity.
In the above Fig. points A, B, and C lie on the same straight line l, so they are collinear points. If a straight line in a plane contains two points but it does not contain the third point, then these three points are said to be non-collinear. In the below Fig. A, B, C are non-collinear.