Rutgers/Lucent ALLIES IN TEACHING MATHEMATICS AND TECHNOLOGY Grant 2000
Using technology not simply to do things better, but to do better things.

USING THE GEOMETER'S SKETCHPAD TO CONSTRUCT BASIC POLYGONS

Defining Transformations

A transformation is an operation that moves a point or figure in a particular way. There are three basic transformations that moves a figure's position without changing its size or shape. These are called isomorphic transformations (iso = same; morph = shape). Each of the three basic operations is described below with a correct mathematical name, a nickname, and an associated "tool".

These basic transformations can be used to create many basic polygons. For example, we can make a square starting with an isosceles right triangle in a variety of ways:

It is the simplicity and symmetrical nature of the regular polygons that makes them so compelling and so important in both the arts and the sciences. Many of them can be constructed directly using compass and straight-edge, as well as using combinations of transformations. Some examples of construction strategies are given in the following sections.

Regular Triangle

Transformational Construction:

Euclidean Construction:

Square

Transformational Construction:

Euclidean Construction:

Other Regular Polygons

Can you describe at least one transformational and one Euclidean construction strategy for the regular pentagon? regular octagon?

Are there patterns governing these strategies?

Parallelogram

Transformational Construction:

Euclidean Construction:

Other Types of Polygons

Can you describe at least one transformational and one Euclidean construction strategy for:

… an isosceles triangle?
… right triangles?
… a rhombus?
… a kite?
… an equilateral but non-regular pentagon?
… an equiangular but non-regular hexagon?
… a figure that exhibits rotational symmetry but not line symmetry?
… a figure that exhibits line symmetry but not rotational symmetry?

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