An Example Activity from

Transforming Geometry with the

University of Georgia

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?