In geometry, a point is a fundamental unit without any quantitative attributes such as size: width, length, height and depth etc. The point only represents a position or location of a variable in plane marked by a dot (.) and is preferably denoted with a capital letter (P, Q, A etc.).
Line or Line Segment
Only when point is extended infinitely or indefinitely in two directions, then point is termed as line or more precisely line segment. A line has infinite length without any width and height i.e., a line is one-dimensional. In 2-dimensional Euclidean geometry, a point is represented by a pair of numbers which is always ordered e.g., (x,y). Here x represents the horizontal, whereas y indicates the vertical component of a plane. In 3-dimensional space, a point is represented by an order of triplet e.g., (x,y,z) in which the third component is added to the space in which a point is located.
For better understanding, the real-life examples of a point can be a star in the sky, the pointed end of a needle, or tip of a ball point etc.
Dimensions of a Point
As discussed above, a point is free of any dimension as it doesn’t own any measurable property. A point itself is zero dimensional and can possess quantitative property only when it is extended in at least one direction; thereby producing a line. One value is required in order to identify the point on line which will provide one dimension or making a line one dimensional.
Similarly, if point is extended to two directions, two values are required to find the point on plane; which creates two dimensions (2D) as seen in circle, square etc. Likewise, if a line is stretched to entirely different direction other than horizontal or vertical components, a new third value is identifiable to find the location of point, giving it three dimensions (3D) as found in sphere, cube etc.
For better understanding, the real-life examples of 2D figures are chessboard, a piece of paper or a pizza. The 3D figures could be basketball, cylinder or an egg etc.
Type of Points
A point can be classified into a few types which are as explained below:
- Collinear Points
- Non-collinear Points
- Concurrent Points
- Coplanar Points
- Non-coplanar Points
- Equidistant Points
The word ‘collinear point’ derived from the word ‘col’ (together) and ‘linear’ (straight line) which means the points which are occurring together in a linear way.
For Example. There are four points D, E, F, G, occurring on a same straight line.
For better understanding, the real-life examples of collinear points are numbers on a ruler, birds sitting in a row on a wire etc.
In contrast to the above type, non-collinear points are those which are not occurring together on a straight line. It can also be understood with the concept; as if a straight line cannot be drawn through a few points, then points are non-collinear in nature.
For Example. There are four points D, E, F, G, not occurring on a same straight line.
For better understanding, the real-life examples of non-collinear points are sprinklers on a cake, coffee beans in a jar i.e., scattered positions of points.
A concurrent point is that point on which two or more lines are intersected.
For Example. Consider the points, D, E, F, and G, making line DE and FG, such that these intersect at point H. Then the point H is known as point of intersection or concurrent point.
For better understanding, the real-life examples of concurrent points are intersecting points of a kite, spider web, spokes of bicycle etc.
The word ‘coplanar’ is derived from the word ‘co’ and ‘plane’ which means, the points which lie on the single plane. In geometry, a plane is known as a 2-D flat surface which extends indefinitely, on which a point, line and three dimensions can lie. In real life, a wall or a piece of paper are the examples of plane.
- There should be at least three points on a plane to be called as coplanar.
For Example. There are four points, D, E, F and G lying on a common plane.
For better understanding, the real-life examples of coplanar points are the hands-on of clock, objects on a table, grids of a graph paper all lie on a single plane.
Unlike coplanar points, the non-coplanar points are those which do not lie on the same plane. There should be at least four or more points which could be called as non-planar only if these are not lying on a single plane. Usually non-coplanar points exist in case of three-dimensional figures.
For Example. There are four points, D, E, and F lying on a common plane, However, the point is said to be non-coplanar as it does not lie on a single plane with others.
For better understanding, the real-life example is the point G on a wall with plane A would be non-coplanar to the points E and F on a wall with plane B.
A point is said to be equidistant from others if it is at similar distance from other points.
For Example. The points E in the given triangle is known as equidistant point which is at similar distance from F and G respectively.
For better understanding, the real-life example of equidistant point is the central point of a sky wheel is equidistant from all other points.