A number of investigations can be done involving quadrilaterals. One investigation that can be explored deals with drawing an original quadrilateral. Then by marking the midpoints of the sides and connecting the midpoints with line segments to create an inscribed quadrilateral.
This can be done using GSP. An example can be seen in the sketch below.
One idea that can be explored is if the shape of the new quadrilateral depends on the shape of the original quadrilateral. One could determine if the two quadrilaterals will have the same shape or different shapes. This could be explored using GSP. In GSP, one could drag the different vertices of the original quadrilateral to form different shapes.
Examples of this using GSP are shown below.
The image above seems to show that if the original quadrilateral is a trapezoid, the smaller quadrilateral will be a rectangle.
The image above seems to show that is the original figure is a rhombus, the second quadrilateral will again be a rectangle.
Another idea that could be explored is if the four smaller triangles created near the vertices of the original quadrilateral have the same area. This can be explored using the measure and calculate features of GSP. An example of this can be seen below.
The calculations show that the four small triangles do not have the same area. However, the calculations do show that the sum of the areas of the four triangles is equal to the area of the smaller quadrilateral.
GSP could be used to explore a number of different aspects of this problem. Examples would include, is there a constant ratio between the perimeter of the smaller quadrilateral and the larger quadrilateral. Another investigation could be done comparing the areas of the two quadrilaterals.
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