**EMAT
6700**

**by
Brad Simmons**

**Median of a Trapezoid**

Use geometer's sketchpad to construct the median of a trapezoid.

Start by constructing trapezoid ZOID. Let segment ZO be a base of the trapezoid and segment ID be the other base.

1. Construct segment ID. Construct a parallel line above segment ID.

2. Construct a point on the parallel line and label it point Z. Construct a segment AB that is the same length as segment ZO (over to the side on the sketch not close to the parallel line). Select point Z and segment AB. Construct circle by center and radius.

3. Construct a point of intersection of circle Z and the parallel line. Label the point of intersection point O.

4. Select the parallel line. Hide the parallel line. Select circle Z. Hide the circle. Select point Z and point O. Construct segment ZO. Select point Z and point D. Construct segment ZD. Select point O and point I. Construct segment OI.

Since segment ZO and segment ID are parallel and segment OI and segment ZD are not parallel, then quadrilateral ZOID is a trapezoid.

**Construct
the median of trapezoid ZOID.**

1. Select segment ZD. Construct the midpoint and label it point M. Select segment OI. Construct the midpoint and label it point N.

2. Select point M and point N. Construct segment MN.

Segment MN is the median of trapezoid ZOID. The median has a special relationship to the two bases. Use the geometer's sketchpad's measure utility to measure segment ZO, segment ID, and segment MN. What relationship do they have? Is the median parallel to the two bases?

Please click here for a geometer's sketchpad sketch that can be manipulated.