WinGeom Investigations

The power of a dynamic geometry program is that a construction can be made (a perpendicular bisector, for example) and you can move ("drag") the points, but the relationships created by the construction (perpendicularity) are maintained.
Note: If constructed correctly, there should be "free" points (e.g. for #1, the four vertices of the original quadrilateral) which you can drag. When these points are dragged, the properties of the construction should remain. Usually, if you attempt to drag constructed points, the entire figure moves.

Directions: Using Wingeom or CabriJr do the following: (Each person will do one. See also below for what to turn in.)

0. Example: Construct a kite. A kite is a quadrilateral with two pairs of consecutive sides congruent. One good way to do the construction is using two intersecting circles. Find properties of the angle measures and the diagonals.
1. Construct a random quadrilateral. Construct the midpoints of each side. Connect these midpoints to make a new quadrilateral. What properties does this new quadrilateral have?
2. Construct a random quadrilateral. Construct the midpoints of each side. Connect these midpoints to make a new quadrilateral. Find the area of the new quadrilateral. How are the areas of the quadrilaterals related?
3. Construct an isosceles trapezoid. Construct the midpoints of each side. Connect these midpoints to make a new quadrilateral. What properties does this new quadrilateral have?
4. Construct a circle. Put 4 points on the circle. Construct the quadrilateral determined by these 4 points. Construct the diagonals - and find the point of intersection. The two diagonals now make four "shorter" segments. Find some relationships between the lengths of these four segments.
5. Construct a circle. Put 4 points on the circle. Construct the quadrilateral determined by these 4 points. Construct the diagonals - and find the point of intersection. The two diagonals now make four "shorter" segments.
   1. When will the four segments be equal?
   2. When will the two segments on one diagonal be equal (but the segments on the other diagonal be unequal)?
   3. When will two segments on different diagonals be equal (and will the other segments be equal or unequal)?
6. Construct various convex polygons (n-gons--not necessarily regular)). Find the sum of the interior angles. Find the sum of the exterior angles (one exterior angle per vertex). Generalize:
   1. For a convex n-gon, what is the sum of the interior angles?
   2. For a convex n-gon, what is the sum of the exterior angles?

What to Turn In

If you turn in a Wingeom sketch:

• On the sketch, have a text box in which you type up your conclusions (save the sketch to your hard drive).
• Submit (e-mail) the wingeom file. (It should have a .wg2 suffix.)

If you "turn in" CabriJr:

• Use TI-Connect to get (at least) three (3) screen dumps which show your properties.
• Create a Word document in which you paste the screen dumps and in which you type up your conclusions.
• Submit (e-mail) the Word file.

Comment on Collaboration

If someone else in the class is doing the same task as you are, each of you should create the sketch and draw your own conclusions first. Once you have done that individually, you may collaborate and share ideas. Each of you should submit your own work.