Points, Lines and Curves

 

Objective:


Geometry has a long and rich history. The term 'Geometry' is derived from the Greek word 'Geometron'. Geometry is the branch of mathematics which deals with the measurement, properties and relationships of points, lines, angles, surfaces and solids. So here we are going to learn about,


1. Points
2. Line segment
3. Line
4. Ray
5. Plane
6. Intersecting Lines
7. Parallel Lines
8. Curves

 

 

Points


The most basic shape in geometry is the Point. In geometry, dots are used to represent points. A point is used to represent any specific location or position. A point can be denoted by a capital letter of the English alphabet. Points can be joined in different ways.


A point has no dimensions such as length, breadth or thickness. A star in the sky gives us an idea of point. Similarly some other examples of points are: the tip of a compass, the sharpened end of a pencil, the pointed end of a needle.
These points will be read as point A, point B point C and point D.

 


Line Segment


A line segment is defined as the shortest distance between two points. These points are called the end points. A line segment is made up of unlimited points. Some examples of a line segment are: an edge of a box, a tube light, and the edge of a post card.

 

For example, if we mark any two points A and B on a sheet of paper, then the shortest way to join A to B is a line segment. It is denoted by. The points A and B are called the end points of the segment.

 Line


A line segment is a part of a line. When a line segment from A to B (i.e. AB) is extended beyond A in one direction and beyond B in the other direction without any end you get a model for a line. A line has arrows at both ends as it can extend indefinitely in both directions. Line is made up of an infinite number of points.


Two lines can either meet at one point only or they will not meet at all. A line can also be represented by small letters of the English alphabet. For example, if a line segment from M to N is extended beyond M in one direction and beyond N in the other, then we get a line, MN. It is denoted by

 


Ray


A ray is a portion of a line. It starts at one point (called starting point) and goes endlessly in one direction. Examples of a ray are: Beam of light from a light house, ray of light from a torch, sun rays. A ray is named using two capital letters. The first capital letter is the starting point of the ray and the second capital letter tells the direction in which the ray is moving.

For example, if a line from M to N is extended endlessly in the direction of N, then we get a ray, MN. It is denoted byand can be read as ray MN.

 


Plane


A plane is said to be a very thin flat surface that does not have any thickness, and is limitless. For example, this sheet is said to plane PQR. An infinite number of points can be contained within a plane. 

 

 

 

Intersecting Lines


If two lines pass through a point, then we say that the two lines intersect at that point. If two lines have one common point, they are called intersecting lines. More than two lines can also intersect at one point. Examples of intersecting lines are: two adjacent edges of your notebook, the letter X of the English alphabet, crossing-roads.


For example, two lines pass though point P. These two lines are called intersecting lines.

 

 


Parallel Lines


Line segments which will not meet, however far they are extended are called parallel lines or non- intersecting lines. Parallel lines never meet, cut or cross each other. Examples of parallel lines are: the opposite edges of ruler (scale), the cross-bars of a window, the lines on a page of the notebook, rail lines.

In the figure, it can be observed that two lines are parallel. We write.

 


Curves

 

Curves can be defined as figures that flow smoothly without a break. A line is also a curve, and is called a straight curve. Curves that do not intersect themselves are called simple curves. The end points join to enclose an area. Such curves are called closed curves.

 

 

For example, Fig (i), (ii), (v) and (vi) are simple curves, whereas (iii) and (vii) are closed curves.

 

Position in a closed curve

A court line in a tennis court divides it into three parts : inside the line, on the
line and outside the line. You cannot enter inside without crossing the line.

In a closed curve, thus, there are three parts.
 (i) interior (‘inside’) of the curve
 (ii) boundary (‘on’) of the curve and
(iii) exterior (‘outside’) of the curve

 The interior of a curve together with its boundary is called its “region”. 

From the below figures lets find out the position of the point P with respect to the circle.

  • Interior of the curve. Here, point P lies inside the circle.

 

 

  • Boundary of the curve. Here, point P is on the circle.

 

 

  • Exterior of the curve. Here, point P lies outside the circle.

 

 

Summary 

 

  1.  A point determines a location. It is usually denoted by a capital letter.
  2. A line segment corresponds to the shortest distance between two points.
  3. A line is obtained when a line segment like AB is extended on both sides Indefinitely.
  4. Two distinct lines meeting at a point are called intersecting lines.
  5. Two lines in a plane are said to be parallel if they do not meet.
  6. A ray is a portion of line starting at a point and going in one direction endlessly.
  7. Any drawing (straight or non-straight) done without lifting the pencil may be called a curve. In this sense, a line is also a curve.
  8. A simple curve is one that does not cross itself.
  9. A curve is said to be closed if its ends are joined; otherwise it is said to be open.

 

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