Trapezoid Area Question

Suppose ABCD is a trapezoid with parallel sides AB and CD. Let M and N be the midpoints of BC and DA.  Then if we know that the distance from line AB to line CD is d and also that the length of MN is k, is this enough to compute the area of the trapezoid.  And if so, what is the area?

Supporting Activity on Paper

1.   Take a piece of the adding machine tape.  Fold it lengthwise so that the fold is parallel to and equidistant from the edges. 

2.   Measure and draw a segment MN of length 4 inches on the fold. 

3.   Then draw and make cuts across the tape to form a trapezoid ABCD so that M and N are midpoints of BC and DA. 

• What is the area of the trapezoid? 
• Can you cut it or duplicate it to make clear the relationship between MN and area that goes beyond simple numerical measurement?

Supporting Activity with GSP

1. Construct a line segment MN. For a point A, construct a line m through A parallel to MN. Construct a point B on this line.
2. Reflect the line m across MN to get line m'. Construct two points C and D on this line m'.

Draw segments to form the quadrilateral ABCD. This is a trapezoid.

• What is the area of the trapezoid? How is it related to the height and to MN?

Can you cut it or duplicate it to make clear the relationship between MN and area that goes beyond simple numerical measurement? Here is one suggestion:

• Construct another congruent copy of ABCD with a common side, so the new one will be CBEF. What shape is AEFD? What is the length of the base of this quadrilateral? How is it related to MN?