6th Grade Math Textbook, Fundamentals

What if you start with a quadrilateral that

has no parallel or congruent sides? Try it.

Make the quadrilaterals as crazy as you like.

Some examples are shown at the right. Be

sure to try some concave quadrilaterals

(quadrilaterals with a “dent,” such as the

one at the far right.)

You should have discovered that, no matter what quadrilateral you start with,

the midpoint quadrilateral is a parallelogram. Why does this happen?

One way to see why is to draw the diagonals of each of the original quadrilaterals

you drew. Here are some examples.

In each case, you should find that two sides of the midpoint quadrilateral are

parallel to one of the diagonals and two are parallel to the other. So, in each

case, the midpoint quadrilateral has two parallel sides. That is, the midpoint

quadrilateral is a parallelogram.

Chapter 9 269

More Enrichment Topics

Use your drawings from this lesson (or make more drawings) to answer
the following questions.

1.If the diagonals of the original quadrilateral are the same length,

what kind of midpoint quadrilateral is formed?

2.If the diagonals of the original quadrilateral are perpendicular,

what kind of midpoint quadrilateral is formed?

3.If the diagonals of the original quadrilateral are the same length

andperpendicular, what kind of midpoint quadrilateral is formed?

4.Discuss and Write Find the midpoints of one (or more) of your midpoint

quadrilaterals. Join them to form a quadrilateral. Tell how the new midpoint

quadrilateral relates to the original quadrilateral.

pages 299–300 for exercise sets.