Three geometrical terms, namely, point, line and plane, form the foundation of geometry.
A solid has a surface which may be flat or curved.
For example, the surface of a box is flat and the surface of a ball is curved. Flat surfaces are known as plane surfaces.
In mathematics, a smooth flat surface which extends endlessly In all the directions is called a plane. A plane has no boundary.
The surface of a smooth wall, the surface of the top of a table, the surface of a smooth black board, the surface of a sheet of paper, the surface of calm water in a pool are all examples of a portion of a plane.
We draw figures such as a triangle, a rectangle, a circle, etc., in a plane. We call them plane figures.
A point is a mark of position. A small dot made by a sharp pencil on a plane paper represents a point.
We name a point by a capital letter of the English alphabet. A point has no length, breadth or thickness.
Let A and B be two points on a plane. Then, the straight path from A to B is called the line segment AB.
Thus, a line segment has a definite length, which can be measured.
The line segment AB is the same thing as the line segment BA.
Measuring Line Segments:
For measuring the length of a line segment, use a 15 centimetre scale. Place the scale along the segment in such a way that the zero mark of the scale matches one end of the segment. Then, read the marking of the scale corresponding to the other end of the segment.
Suppose, the scale reads 3 centimetre, the length of the segment is 3 centimetre.
1) Let us find the length of the given line segment AB:
Place the 15 centimetre scale along the line segment AB such that the point A coincides with the zero mark of the scale. Observe that the other end B of the Line segment AB reads 7 cm. So, the length of the line segment AB is 7 cm.
The length of a segment may not be equal to exact number of centimetres. In that case, read the smaller divisions further which show millimetres.
2) Let us find the length of the given line segment CD:
Place the 15 centimetre scale along the line segment CD such that point C coincides with the zero mark of the scale. Observe that the point D reads 9 centimetres 5 millimetres. So, the length of the line segment CD is 9 cm 5 mm. In short, it is written as l(CD) = 9 cm 5 mm.
A line segment extended endlessly in one direction is called a ray.
Thus, a line segment AB, extended endlessly in the direction from A to B, is a ray. The ray AB has one end point, namely A, called its initial point.
Clearly, a ray has no definite length.
BA is a ray with initial point B and extending endlessly in the direction from B to A.
Clearly, AB and BA are two different rays. An unlimited number of rays can be drawn in different directions with a given point O as the initial point.
A line segment extended endlessly on both sides is called a line.
Thus, a line segment AB extended on both sides and marked by arrows at the two ends, represents a line, denoted by AB or BA.
Sometimes, we represent a line by a small letter l, m, n, etc.
Two intersecting planes intersect in a line. A line has no end points.
Result 1) An unlimited number of lines can be drawn passing through a given point.
In the above figure, lines m, n, p all pass through a given point O.
Result 2) If two different points A and B are given in a plane then exactly one line can be drawn passing through these points.
If there is a point P common to two lines l and m, we say that the two lines intersect at the point P and this point P is called the point of intersection of the given lines.
If no point is common to two given lines, it would mean that the lines do not intersect. Such lines are known as parallel lines.
The rails of a railway line, opposite edges of a ruler and the opposite sides of a rectangle are examples of parallel lines.
It is clear that either one point is common to two given lines or no point is common to them.
Result 3) Two lines in a plane either intersect at exactly one point or are parallel.
Three or more lines in a plane are said to be concurrent if all of them pass through the same point and this point is called the point of concurrence of
the given lines.
In the above figure, the lines m, n, p are concurrent lines, since all these lines pass through the same point O.
Three or more points in a plane are said to be collinear if they all lie on the same line and this line is called the line of collinearity for the given points.
The three points A, B, C are collinear.
Now, the points A, B and C are non collinear.
Distinction between a Line Segment, a Ray and a Line:
|1) A line segment has two end points.
2) A line segment has a definite length.
3) A line segment can be drawn on a paper.
|1) A ray has only one end point.
2) A ray doesn’t have a definite length.
3) We can’t draw a ray on a paper. We can represent it by a diagram.
|1) A line has no end point.
2) A line doesn’t have a definite length.
3) We can’t draw a line on a paper. We can represent it by a diagram.