6th Grade Math Textbook, Fundamentals

Enrichment:

Quadrilaterals from Quadrilaterals

Objective To investigate the nature of quadrilaterals formed by joining the midpoints of other quadrilaterals

• To make conjectures about the conditions needed to form a particular quadrilateral

As you know, a square is a quadrilateral with four right angles and four
congruent sides. What kind of quadrilateral would you create if you
connected the midpoints of the sides of a square?

Use grid paper and a ruler.

Draw a square. Find the midpoints of its sides and join them.

You can see that the new quadrilateral that is formed is
another square. Compare drawings with other students
to be sure that you all have the same results.

A rectangle is a parallelogram with four right angles.

What kind of quadrilateral is formed when you connect

the midpoints of the sides of a rectangle?

Draw a rectangle that is not a square and join the

midpoints of its sides.

You can see that a rhombus is formed. Compare drawings

with other students to check that everyone has drawn

a rhombus.

Now, try connecting the midpoints of the sides of a

non-rectangular parallelogram, a trapezoid, and a kite.

Your shapes should look different than the ones below.

Parallelogram Trapezoid Kite

So far, for all the quadrilaterals you have tried, you should have found that

the “midpoint quadrilateral” is a parallelogram. (Sometimes it is a square

or a rectangle or a rhombus, but these are all parallelograms.)

Does this surprise you? Maybe not. After all, each of these quadrilaterals is

“regular” in some way. For example, a parallelogram has two pairs of sides

that are parallel and congruent, a trapezoid has one pair of parallel sides,

and a kite has two pairs of sides that are the same length.

268 Chapter 9