Title
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 nonconvex?.
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.
jtaylor1@rockdale.k12.ga.us
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