USING THE GEOMETER'S SKETCHPAD TO EXPLORE MIDPOINT QUADRILATERALS
I will begin by assuming that most of you have had some exposure to the Geometer's Sketchpad (GSP) before. As a review, show that you can:
- create a non-special quadrilateral;
- change the labels of the vertices;
- measure its angles, and show that the sum of the angles remain constant even as the shape of the figure is dynamically changed;
- construct the interior of the figure, color it red, and measure the area;
- display a table that shows the perimeter and area for different quadrilaterals.
To illustrate how the Sketchpad can be used to explore geometric relationships, follow the sequence of steps outlined below:
- Begin with a new sketch. In Display >> Preferences, turn off Autoshow Labels for all objects.
- Construct another non-special quadrilateral.
- Construct the midpoints of all of the sides; connect the midpoints to form a "midpoint quadrilateral". Observe the shape of the midpoint quadrilateral as, in turn, you grab-&-drag each of the vertices.
- Make as many conjectures as you can about this inner figure.
- Describe how you could confirm or deny the correctness of each conjecture.
For example, one conjecture is that the inner figure is a parallelogram. You could confirm or deny that by measuring the slopes of the sides of both figures. If it's true, what reasonable explanation might there be for this?
Exploration: Assuming the inner figure is always a parallelogram, under what conditions is the inner figure a rectangle? a rhombus? a square?