Essence of Geometrical Constructions
What is a geometrical construction? A simple construction is that of a line. By moving a sharp-tipped pencil against a straight-edged ruler, you can draw a straight line. But as we discussed earlier, such a line will only be an approximation to the actual concept of a line. The ideal, perfect line has zero width and extends to infinity on both sides. The line you have drawn using a straight-edged ruler and a pencil will have some finite non-zero width (you can confirm this using a microscope) and it will also not extend for infinite lengths.
On the other hand, suppose that you try to draw a straight line using just a pencil (no ruler). No matter how steady your hand is, the “straightness” of your line will be poorer than the line you can draw with the help of a ruler. In geometrical constructions, we always use a ruler to draw straight lines, because a ruler helps us draw lines which are close to the ideal concept of a line.
The essence of geometrical constructions is that whenever we are carrying out a construction, we are thinking of ideal geometrical concepts in our mind, even though what we are able to actually draw is only an approximation to those ideal concepts. Thus, when we mark a point using a sharp-tipped pencil on a paper, we are thinking of the ideal concept of a point. When we draw an angle of 900, we are thinking of an angle of exactly 900 formed between two lines of exactly zero width – even though what we are able to draw on paper would be an angle very close to 900 between two lines of finite, non-zero width. No construction can be perfect, but we can come close to the ideal concepts of geometry if we do the construction properly.
For geometrical constructions which require no measurements of lengths and angles, only two instruments are to be used: an ungraduated ruler, which is also called a straight-edge (a ruler with no length markings), and a compass. Using only these two instruments, many beautiful geometrical constructions can be done. When measurements of lengths or angles are required, a graduated ruler and a protractor can be used.