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.

 

  1. 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.