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Quadrilaterals - Definitions and constructions

Personal definitions of a square

In our class, we discussed your "personal" definitions of a square. Two important take-away messages from this discussion are:
  1. There are many ways of defining the same object. We need to consider our starting point - Do we already know what a rectangle is? Do we know what a regular polygons is? Or perhaps we only know that square is a quadrilateral?
  2. Correct definitions may still contain redundant references. For example, defining square as a "regular polygon with four congruent sides and four congruent angles" is correct, although we don't have to list all of these conditions. (Which ones can be dropped?)
We will call definitions that have no redundant conditions minimal definitions. __________________________________________________________________________________________

Minimal definitions of quadrilaterals

In order to agree on what different quadrilaterals are, we need to agree on their definitions. The following is the list of minimal definitions of quadrilaterals as we are going to use them. Please note the definition of a kite, trapezoid and square - is there something unexpected there? Quadrilateral - a four-sided polygon Kite – quadrilateral with two distinct pairs of adjacent congruent sides Trapezoid - a quadrilateral with at least one pair of parallel sides Isosceles trapezoid - a trapezoid with congruent base angles Parallelogram – a quadrilateral with two pairs of parallel sides Rhombus – a quadrilateral with four congruent sides Rectangle – a quadrilateral with three right angles Square – a quadrilateral with four congruent sides and one right angle ________________________________________________________________________________________

Constructing quadrilaterals

Knowing the defitions of quadrilaterals, we can now try to construct them. As you will see, construction of quadrilaterals can be quite a challenging task as we are trying to come up with the most general construction for each quadrilateral. For example, if you are constructing a rhombus, you should be able to manipulate your construction (drag vertices) to create any rhombus you want but no quadrilateral that is not a rhombus. Sometimes you'll see that your construction "breaks down" and you end up with a line or no shape at all. This should be fine as long as you are able to manipulate your quadrilateral and:
  1. Create any quadrilateral of the chosen category (kite, rhombus, parallelogram,...).
  2. Whenever your shape is a quadrilateral, it is a quadrilateral of that category.
We will talk about these constructions in great detail as they bring up important questions and deepen our understanding of quadrilaterals. So let's get started! Note: We will not have time in class to discuss all constructions but try your best to construct them all. We will be discussing them in the order in which they are listed here, so work from #1. You may open your Geogebra so that you can save your work or if you are working here, make sure to download your creations by going to the sandwich menu > Download as... > .ggb . If you are logged in to your geogebra account, you can bypass download and use "Save" to save your creations directly to the GeoGebra cloud.)

1. Construct a rhombus

2. Construct a kite

3. Construct a trapezoid

4. Construct a parallelogram

5. Construct a rectangle

6. Construct a square

7. Construct an isosceles trapezoid