Ruler and Compass

Mathematics as a science is built as an axiomatic theory. Some statements that are self-evident, are assumed to be true; such statements are called axioms. Every statement that is not an axiom must be proven. When proving a statement, you are allowed to use axioms and other statements that have been proven. A statement that has been proven is called a theorem.

In geometry proofs, phrases like "...draw a line perpendicular to..." are common. In order to prove that you can draw a perpendicular line, you use a so called ruler-and-compass construction; you then have to prove that your construction is correct. Once a construction has been proven you can use it in proofs or in other constructions.

The ruler and the compass are assumed to have no markings, they can hence not be used to measure distances in terms of units. In order to handle distances you must use the fact that the distance between a point on the circle and the midpoint is the same for all points on the circle.

Using dynamic software it is easy to check whether a construction is correct or not, you just drag some points.

Demonstration

When you make a ruler and compass construction in GeoGebra, you may only use tools that corresponds to a ruler and a compass without any markings. You can, for instance, restrict the set of tools to:

Icon Circle, Icon Intersect Two Objects, Icon Line, and Icon Segment

This is a demonstration of how to solve the task in the box below.

Start with: A line segment.

Construct: A perpendicular bisector.

 

Demo of construction protocol. Click on Play to see how the construction is made.

If you want to show how a construction was made, you can choose Navigation Bar for Construction Steps under the View-menu. Below the drawing pad you will see the navigation bar.


Exercises

No one knows for certain why the circle is divided into 360°. One theory is that it has to do with ancient Babylonia where they used the base 60. You may wonder how the base 60 is related to the number 360 when it comes to circles; the exercises below will demonstrate how the number 360 appears when using the base 60 and making one revolution.

In order to get an understanding of circle patterns, you can start by doing the exercises Geometry - Dynamic Patterns.

A restricted set of tools used to make ruler and compass constructions.

Exercise 1 - Isosceles triangle

Construct an isosceles triangle as a ruler-and-compass construction. When you are done hide all auxiliary objects.

Exercise 2 - Equilateral triangle

Construct an equilateral triangle as a ruler-and-compass construction.

Note that you can make circles through existing points.

When you are done, hide all auxiliary objects. You should be able to move all movable points and the triangle should remain equilateral.

Exercise 3 - Regular hexagon

Image

Make a regular hexagon as a ruler-and-compass construction!

further info:

axiom: http://en.wikipedia.org/wiki/Axiom

ruler-and-compass construction: http://en.wikipedia.org/wiki/Compass_and_straightedge_constructions

about degrees: http://en.wikipedia.org/wiki/Degree_%28angle%29

by Malin Christersson under a Creative Commons Attribution-Noncommercial-Share Alike 2.5 Sweden License

www.malinc.se