In Chapter 7, you discovered that the midsegment of a triangle is not only parallel to the third side, but also half its length. But what about the midsegment of a trapezoid?

The diagram at right shows a midsegment of a trapezoid. That is, $\overline{EF}$ is a midsegment because points $E$ and $F$ are both midpoints of the non-base sides of trapezoid $ABCD$.  

1. If $A\left(0,0\right)$, $B\left(9,0\right)$, $\ C\left(5,6\right)$, and $D\left(2,6\right)$, find the coordinates of points $E$ and $F$. Then compare the lengths of the bases ($\overline {AB}$ and $\overline {CD}$) with the length of the midsegment $\overline{EF}$. What seems to be the relationship?

   Hint (a):

   How do you find the lengths from coordinates?
   Think of how the answers could be related.

   Answer (a):

   It seems to be the average of $AB$ and $CD$.

2. See if the relationship you observed in part (a) holds if $A\left(-4,0\right)$, $B\left(2,0\right)$, $C\left(0,2\right)$, and $D\left(-2,2\right)$.

   Hint (b):

   Use the same method you used to solve part (a).

3. Write a conjecture about the midsegment of a trapezoid.

   Answer (c):

   The midsegment of a trapezoid is parallel to the bases and has a length that is the average of the lengths of the bases.