Habits of Mind Problem #5

Exploring Quadrilaterals with GeoGebra

Due Wednesday, September 26

Consider any quadrilateral ABCD. Suppose a second quadrilateral Q is formed using the midpoints of the segments AB, BC, CD, and DA as its vertices. In this HoM you will investigate what has to be true of the original quadrilateral ABCD, for the quadrilateral Q to have certain properties.

1)Basic set-up using GeoGebra

  1. Use the segment or polygon feature to make a quadrilateral P with vertices (0,0), (1,3), (5,5), and (7,1).
  2. Use the midpoint feature to add the midpoint of each edge of P.
  3. Use the polygon feature to create a new quadrilateral Q by selecting the four points created in the previous step as the vertices of Q.
  4. Use the segment feature to add the diagonals of Q, and add the intersection point using the “intersect two objects” feature.
  5. Use the segment feature to add the diagonals of P and add the intersection point.
  6. Use the angle feature to find one of the angles formed by the diagonals of Q at their intersection.
  7. Use the angle feature to find one of the angles formed by the diagonals of P at their intersection.
  8. Use the angle feature to calculate one of the interior angles of Q.
  9. By using the move feature (the arrow on the top left of the GeoGebra toolbar) you can move your angle labels and other labels so that the figure looks neat and easy to read. Print your figure (either from GeoGebra or copying and pasting it in a word file).
  10. Print the Construction protocol window.

2)Exploration: Use the move feature to dynamically move the vertices of the quadrilateral P around, and see how the features of Q change. Use your observations and your knowledge of quadrilaterals to briefly answer the following questions.

  1. When both P and Q are parallelograms, what do you observe about their diagonals?
  2. When Q is a rectangle, what do you notice about the diagonals of P?
  3. What type of quadrilateral should P be so that Q is a parallelogram?
  4. Print one example figure for each of the above questions. (The construction protocol window is not necessary for these.)

Note: Recall that you can go to “View” and choose whether to have the grid or axes showing. Also, you can go to “options” and then “Labeling” and select “no new objects” before you begin if you don’t want your figure too cluttered. By clicking on the circle next to each object in the algebra window, you can also hide or show the object, accordingly. If at any time you make a mistake in GeoGebra, just click the yellow arrow on the top right to undo the current or most recent action.