Balancing Triangles and Quadrilaterals
An Example Activity from 
Transforming Geometry with the Geometer's Sketchpad

Dr. John Olive
University of Georgia

[Instructor's Note: This activity comes from an exploration I did with my second and third grade class in Atlanta many years ago. It helped my students develop an intuitive feel for balance in two dimensions. It has proved to be a challenging problem for mathematics teachers and students of all ages! You will need to prepare enough cardboard triangles and quadrilaterals for each pair of students to have at least one of each. The triangles and quadrilaterals should vary in type. Most should be irregular and convex. Corrugated cardboard boxes are a suitable source for the cardboard that needs to be stiff enough to remain flat when cut. The dimensions should be about 30 cm. by 20 cm. depending on the size of available boxes. The edges of your shapes need to be straight so a large paper cutter is suggested for cutting your shapes.
A fulcrum for balancing the shapes can be made from an old tennis ball sitting in a small paper juice cup. The diameter of the top of the cup needs to be smaller than the diameter of the tennis ball so that the tennis ball protrudes above the top of the cup. The cup needs to have a flat bottom so that it can stand securely on a table top.
Each pair of students will also need sharp pencils and 50 cm. rulers or metre sticks.]

Cardboard triangles on tennis balls
Work with a partner and try to balance your cardboard triangle on the tennis ball. The plane of your triangle should be horizontal. Mark the balance point with your pencil. Can you construct the balance point geometrically? (All you need is a ruler and pencil.)

Finding the balance point of a triangle with GSP
If you came up with a construction for the balance point of a triangle, construct it with GSP. Can you use your Centroid script to find the balance point?

Cardboard quadrilaterals on tennis balls
Do the same experiment with an irregular cardboard quadrilateral. Is it as easy to find the balance point of this figure as with the triangle? Explore with your partner ways to geometrically construct the balance point of your quadrilateral. Discuss your ideas with other pairs of students. Look at the different quadrilaterals and see if your ideas would work for each of them.

Finding the balance point of a quadrilateral with GSP
Find a way to use the Centroid Script for a triangle to construct the balance point of any convex quadrilateral in GSP. Hint: Is a median a balance line of a triangle? If so, why? [Click here for a GSP Solution]
Demonstrate that any line passing through the balance point (center of gravity) of a convex quadrilateral will be a balance line for that quadrilateral.

[The Sketch BalLineQuad.gsp simulates the balance line of a quadrilateral. The measurements and calculations show the relationship of the areas of the different regions formed by the balance line and their moments about the balance line. Change the position of the balance line by rotating it about the center of gravity. What happens to the areas and the moments?]

Challenge: Extend your construction strategy for the balance point of a quadrilateral to find the balance point of any convex pentagon.
[Click here for a GSP Solution]

General Discussion (Whole group)
What construction methods were used to find the balance points of the triangles?
What construction methods were used to find the balance points of the quadrilaterals?
Discuss the ideas that did not work. What assumptions were made? Which assumptions proved to be false? Why?
In what ways does the construction of the Center of Gravity of a triangular region generalize to the construction of the Center of Gravity of a quadrilateral region? In what ways does it not generalize?
Can the construction strategy for the center of gravity of a quadrilateral be generalized for any convex polygon? What about concave quadrilaterals? Will the construction still work?
[Click here for a possible GSP solution]

The Sketch HexagonCG.gsp demonstrates the balance point of a convex hexagon. It was created using the GSP script for the center of gravity of a quadrilateral.

Question to ponder: If the Center of the Data for a set of points in a coordinate plane is defined as the point with coordinates (Mean x, Mean y), where Mean x is the arithmetic mean of the x-coordinates of all the points, and Mean y is the arithmetic mean of all the y-coordinates, will the Center of Data correspond to the Balance Point of the polygon formed by the points?

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