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Constructing an equilateral quadrilateral II

Constructing an equilateral quadrilateral II

Description:

To demonstrate a second rhombus construction with a focus on the geometric relationships inherent in the construction-both those we intend, and those that are an unintended consequence.

This packet follows up on three prior packets introducing compass and straightedge constructions. It demonstrates a construction for a rhombus, in which the original segment is a diagonal of the constructed rhombus, rather than a side.

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Tutorial

Introduction

This packet assumes that you know how to construct an equilateral triangle and that you have worked through a general construction for a rhombus. And of course you need to know why a compass draws a circle.

In this packet you will learn a second way to construct an equilateral quadrilateral (a rhombus). The first construction for a rhombus uses the original segment as a side of the rhombus. In this construction, we will use the original segment as a diagonal.

Don't forget that you want to be thinking about why these constructions work, not just memorizing the steps. If you're only memorizing steps, you're not learning any geometry.

A second general construction

This video demonstrates a second rhombus construction. In this construction, the original segment becomes a diagonal of the rhombus, rather than a side.

Conclusion

This packet, together with my others on geometric constructions, is intended to demonstrate an important habit of mind-looking at constructions as ways of showing and exploring relationships among geometric objects. They are not intended to facilitate memorization of a series of steps.

In that spirit, consider the final image in the video you just watched (see below). The only relationship we constructed is that the sides are all the same length-we constructed an equilateral quadrilateral. But many other properties come along for free. While we only intended to construct equal-length sides, we also got some equal-sized angles. Angle ADB is the same size as angle BCA, even though we didn't try to make it so.

What other relationships appear to be true in this diagram? For any relationships you notice, you should ask two questions:

  1. Are these relationships really true, or do they just appear to be? and
  2. Are these relationships the result of the particular rhombus we constructed, or would the same relationships be true no matter what rhombus we construct?