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Quadrilaterals Inscribed Inside Quadrilaterals

Problem Statement
Given any quadrilateral, construct a midpoint on each side. Connect each consecutive midpoint with a segment. What are the properties of the shape formed by joining the midpoints? Does the resulting shape depend on the type of quadrilateral (e.g. convex & concave)? How does the area of the new figure compare to the area of the original quadrilateral?

Suppose you want the new figure to be a rectangle. What quadrilateral would you start with? What quadrilateral would you start with so that the new figure to be a rhombus? What quadrilateral would you start with so that the new figure is a square?

Will the new figure ever be non-convex?

Problem setup

The problem asked us to create quadrilaterals without any special designation. We were then to construct midpoints on each side and create new quadrilaterals using those midpoints. We were then asked to investigate the properties of the new quadrilaterals. There were also several investigative questions which needed to be answered.


Plans to Solve/Investigate the Problem

I plan to construct both a concave quadrilateral and a convex quadrilateral. I will then construct two more quadrilaterals using the midpoints, calculate lengths, areas, perimeters, and area relationships.


Investigation/Exploration of the Problem

I constructed the quadrilaterals as shown below.



It is my observation that the quadrilaterals constructed, both concave and convex, are parallelograms. It is interesting to note that the areas of the smaller quadrilaterals are exactly ½ of the larger quadrilaterals. I did not, however, find any particular special quadrilaterals who’s midpoint made a square or a triangle. Using Geosketchpad, I was able to construct right angles without any special quadrilateral being drawn. (See sketch below):




Author & Contact
Jim Taylor.


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