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Title
Quadrilaterals 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 mid points? Does the resulting shape depend on the type of quadrilateral? How does the area of the new figure compare to the area of the original quadrilaterals?

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 would be a rhombus? What figure would you start with so that the new figure is a square?

Will the new figure ever be non-convex?

Problem setup
Construct a quadrilateral inside a quadrilateral whose sides meet at the midpoints of the outside quadrilateral. Once the two quadrilaterals are constructed, examine and analyze the properties of the interior quadrilateral as well as how the two quadrilaterals relate and compare to each other. What shape is the interior figure? What shapes would you start with to create a specific figure inside? What figure would you start with to end up with a non-convex figure? This assignment is consistent with our classroom approach. We learn through the investigation of manipulatives. Through this investigation we generate our own beliefs about the relationships between quadrilaterals as well as each of their properties.

Plans to Solve/Investigate the Problem
To investigate the problem I plan to use dynamic geometry software, in this case Geometer’s Sketchpad, to construct, analyze, and measure the individual shapes as well as the relationships between them. Once I have done the constructions and measurements I will compare the data and see if I can make any generalizations. I will also use a paper-pencil sketch along with software to investigate the resulting figures as well as finding what figure I must start with to end up with a specific figure inside. I will also construct as many different quadrilaterals as possible and construct the quadrilateral on the outside to see if the non-convex figure is able to be made.

Investigation/Exploration of the Problem
To solve this problem I began by trying to visualize the properties of a quadrilateral as well as the properties of the resulting figures after construction. While I was visualizing, I tried to recall what I knew about specific quadrilaterals. If you need a reference visit the Intermath dictionary or use a geometry text. I began constructing special quadrilaterals using Geometer’s Sketchpad, including a square, rectangle, rhombus, kite, arrowhead, and parallelogram. Once the quadrilaterals were constructed, I constructed the interior figures according to the investigation instructions. I began to classify them according to their properties, all of which were quadrilaterals. For example, my first construction was a rectangle. I measured the interior figure and found it to be classified by its properties as a rhombus; a parallelogram with four equal sides. I continued this process until I determined what each interior figure would be. After calculating and investigating the properties of the interior figures, I concluded that squares give squares, rectangles give rhombi, rhombi, kites and arrowheads give rectangles, and parallelograms give parallelograms. After determining what the interior figures were, I used Sketchpad to calculate the area of the original and resulting figures. After analyzing the measurements, I determined that every figure was twice the size of the resulting interior figure. I through inductive reasoning I generalized that the interior quadrilaterals are half the area of their original figures. I also observed that none of the resulting figures were non-convex. I started sketching, on paper, every quadrilateral shape I could draw. None of the resulting figures were non-convex. Through inductive reasoning I generalized that the non-convex figure could not be constructed using the instructions given by the investigation.

Extensions of the Problem
1: How does the perimeter of the new figure compare to the perimeter of the new quadrilateral?

2: If you start with a non-convex figure, can you then construct an outside quadrilateral that has the midpoints of its sides as the vertex of the interior non-convex quadrilateral?

I drew an arrowhead and so far I have been unable to construct an outside quadrilateral that satisfies the conditions in extension number two.

Author & Contact
Nathan Little
nlittl@aug.edu

Link(s) to resources, references, lesson plans, and/or other materials
My Constructions
Intermath

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