A Favorite Classic That Needs to be Shared...
Note A: This problem is best explored using dynamic geometry software (e.g. GSP), but it can be explored using pencil-and-paper as well.
Step 1: Draw a quadrilateral of any sort.
Step 2: Bisect each of the four sides.
Step 3: In a clock-wise direction, connect the adjacent midpoints with line segments.
Step 4: What do you notice? That is, make a list of conjectures that appear to be true. Can you prove any of them?
Note B: Make sure you try many different types of quadrilaterals, which is why dynamic geometry software is a helpful tool.
Final Task: Extend the problem by starting with a pentagon, hexagon, ..., n-gon...again of all different types. Do any of your previous conjectures remain "true"?
Hint: When making you conjecture list, focus on things such as shape, lengths, and area?
Solution Commentary: When I first saw that the resultant shape was a parallelogram, I admit to being blown away. Then, when trying to prove it, I smiled, as it became quickly obvious using standard theorems in a high school geometry course. NICE!
As part of your original conjectures, did you make one regarding the perimeter of the parallelogram fomed and the diagonals of the quadrilateral? If not, explroe it.