Symmetry Challenges

In his online article Simply Symmetric, Michael de Villiers observes that symmetry is a powerful but often overlooked tool for formulating proofs:

Most primary geometry curricula around the world introduce the concept of line symmetry fairly early, and sometimes also that of rotational, translational and glide reflective symmetry. However, in high school and even college geometry textbooks, one still tends to find a general lack of using symmetry to not only organize and classify geometric objects, but also to more easily logically explain (prove) certain properties of these objects as theorems.

Symmetry can also play a key role in constructions. In a prior blog post, I challenged you to construct a rhombus in multiple ways by using curated collections of geometric constructions tools. Now you’ll be constructing more polygons, leveraging the symmetry properties of each polygon as you build it.

The nine pages of the Web Sketchpad model below challenge you to construct a specified triangle or quadrilateral with small sets of tools. The tools vary from page to page. On page 6, for example, you’re asked to construct a rhombus with a Reflect Point and Quadrilateral tool, whereas on page 7, you agin build a rhombus, but with a Segment, Rotate, and Quadrilateral tool instead.

To judge if your work is a success, drag the various points in your websketch and notice how your construction behaves. Does your rhombus, for example, change its side lengths and angle measures but still remain a rhombus?

The video at the end of this post demonstrates some of the constructions, but I recommend engaging with them first yourself. If you need a general introduction to Web Sketchpad’s tools, check out our new YouTube collection of Web Sketchpad Help videos.

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