What figure is formed? - December 1995
What figure is formed when the consecutive midpoints of the sides of a quadrilateral are joined? What if the original quadrilateral were a rectangle? A kite? An isosceles trapezoid? A square? A rhombus? Other shapes? Explain why you think your answer is true.
From: Eglsnest@aol.com Special Mention
Students from the Garland Street Middle School, Bangor, Maine.
Investigation on: What figure is formed when the consecutive midpoints of the sides of a quadrilateral are joined?
When the consecutive midpoints of the sides of a quadrilateral are joined, a closed figure is always formed. In this project, we investigated what figure was formed when the midpoints were joined. In the following paragraphs we will discuss our findings and conclusions.
Before we undertook the project we researched the properties of different quadrilaterals, to gain a background of this subject. Then we began our project. We discovered that when the consecutive midpoints were joined in a parallelogram, a smaller parallelogram was formed in the interior of the original parallelogram. The joining of the consecutive midpoints of a trapezoid also formed a parallelogram. The midpoints of an isosceles trapezoid formed a rhombus, as did the midpoints of a rectangle. When the consecutive midpoints of a kite or a rhombus were joined, a rectangle was formed. When the consecutive midpoints of a square were joined, a square was formed.
In any quadrilateral, a parallelogram was formed when we joined the consecutive midpoints. Some figures such as the trapezoid formed shapes in their interior that only had the properties of a parallelogram. But some shapes, such as the square formed special types of parallelograms. Even concave or irregularly shaped quadrilaterals formed parallelograms when their consecutive midpoints were joined. In our attached captioned sketch, we indicate why our theory is always true.
Ashok Surapaneni, age 13, 7th grade
Julie Pancoe, age 13, 8th grade
Aly Theeman, age 13, 8th grade
Claire Gross, age 13, 8th grade
Jessica Rosenblatt, age 13, 8th grade
Garland Street Middle School, Bangor, Maine
Comments
I chose to give this piece a special mention (a new category), because their problem was presented very nicely. They also got the general case and the proof (in the first sketch) and did seven examples. They didn't prove those examples (which I felt put them out of the running for greater glory), but I really like their introductory document, which you see here. It just reads well. They introduce the general case and tell a bit about how they tackled the problem. That gives the reader some insight into what direction the solution might take. They also looked at concave and weird-shaped quadrilaterals, which no one else did. Cool stuff!
Back to December 1995 List of Winners
15 March 1996