Module 13 Unit 1

Part 1: Geometric Construction Basics

What do we mean by geometric constructions?

The first three postulates in Book I of Euclid’s Elements describe which constructions were allowed in his development of geometry;

Postulate 1: A straight line can be drawn from any point to any other point.
Postulate 2: A finite straight line can be produced continuously in a straight line.
Postulate 3: A circle can be described with any center and radius.

For Euclid and the other mathematicians of his time, the construction of the lines in Postulates 1 and 2 were carried out with an unmarked straightedge, while the circle in Postulate 3 was described with a compass. They restricted the use of these tools to the following specific tasks:

Given two points P and Q in the plane, a straightedge can be used to construct the line passing through P and Q.
Given a point P and a line segment RS of length |RS| = r, a compass can be used to construct a circle centered at P of radius r.

A geometric figure F is constructible from a given geometric figure G, if all of the components of F can be obtained from those in G in a finite sequence of constructions of the sort described in 1) and 2) above. We say that the figure F results from the figure G by a geometric construction and the finite sequence of constructions is called a construction algorithm for G given F.

This meaning for the term geometric construction is still current today! Although we now have computer software such as Geometer’s Sketchpad that can replace the compass and straightedge for actually carrying out geometric constructions, such software is designed to produce constructions that are exactly the same as those that can be achieved with a straightedge and compass. Such computer programs also have capabilities to:

Measure angles, lengths, distances, and areas in a figure.
Enhance the display of a geometric figure with shading, color, labels, and explanatory text.
Perform geometric transformations such as rotations, tranlations, dilations and reflections on geometric figures.
Change a geometric construction dynamically, that is, change the positions and distances between components of the given figure while still maintaining the prescribed relationships between constructed components of the figure.

Nevertheless, the geometric constructions that can be produced by these high-tech wonders are, by design, essentially the same in mathematical content as those achievable with a simple compass and straightedge.

Why are geometric constructions important for geometry students and teachers?

Trying to learn geometry without using geometric construction is like trying to learn chemistry or biology without using laboratories. Basic knowledge and skills on geometric constructions help students to discover and explore geometric relationships and interpret geometric concepts and theorems. They can also help the teachers to transform the static and confusing array of definitions and theorems typically found in geometry textbooks into an active and exploratory investigation of geometric relationships. Computer-based geometric construction programs such as Geometers Sketchpad enable both students and teachers to explore geometric relationships dynamically and to create very complex and yet very precise geometric constructions and diagrams.

As any mathematics teacher knows, and as most students soon come to realize, geometric figures are nearly always helpful for the analysis and solution of real world problems and for learning new mathematical ideas. Although the geometric constructions as we have defined them are not, strictly speaking, necessary for such graphical descriptions of problems or ideas, their precision can often reveal aspects of the problem or idea that may not be evident in an informal paper or blackboard sketch. Teachers usually find that extra time spent in producing excellent graphical reprentations and figures for class handouts and exams is rewarded by better student understanding and interpretation of problems and ideas.

1.1.1 Just Do It!