A. Midpoint Quadrilateral of a quadrilateral ABCD.

1. Construct a quadrilateral ABCD and also construct the midpoints of its sides and join them to form the midpoint quadrilateral MNOP of ABCD, as in the figure.



Observe and Conjecture 1 . Drag the points around and observe the midpoint quadrilateral. What kind of quadrilateral do you conjecture it to be?

Observe and Conjecture 2. Look harder as you observe. Is there any relationship between a segment such as MN and any feature of the original ABCD? What is the relation?

Can you prove your conjecture in Conjecture 1? If so, do so. If not, go on to the hint in the next section.

 

2. Draw in the diagonals of ABCD. Now look for relationships between these diagonals and the sides of MNOP.

   Now prove the conjecture about the shape of MNOP.

    

   Also, use the same proof to prove that the perimeter of MNOP is related to something about the diagonals of ABCD.

B. Triangle ABC given the midpoint triangle first.

Construct in the new sketch a triangle ABC and its midpoint quadrilateral XYZ, with X = midpoint BC, Y = midpoint CA, Z = midpoint AB.

This is a familiar figure, but we can look at it in a new way. Instead of saying Z = midpoint AB, we can say the B is the point reflection of A in Z. Here is how to do point reflection in Sketchpad.

• Start in a new figure with a point M and a point A. Select M and choose Mark Center in the Transform menu. Select A and choose Rotate in the Transform menu. Type in 180 degrees in the dialog box for the rotation angle. This constructs a new point A' which is the point reflection of A in M.
• Connect A and A' by a segment and drag the points around to see the relationship. Is it clear why rotation by 180 degrees is the same as point reflection in the plane?
• Now add a point N to the sketch and reflect A' in N to get A''. Also construct segment MN and the triangle AA'A''. This figure is a triangle and a midpoint segment, but it is built from the free points M, N and A, not the triangle AA'A''. Thus the points M and N can be dragged independently.
• From the point of view of motions of the plane, this says a double-point reflection with center M and then N is the same as a translation in the direction of MN but twice the distance. We will do more with this later.

C. Quadrilateral given the midpoint parallelogram.

• In a same sketch, draw another point O and construct the parallelogram MNOP.
• Then reflect A'' in O to get A''' and construct the segments to give quadrilateral AA'A''A'''. How is MNOP related to this parallelogram?

• Now based on this, state clearly how one can start with a parallelogram MNOP and any point A, how to construct a quadrilateral with MNOP as the midpoint quadrilateral and with A as a vertex.