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### Why do the Midpoints of Quadrilaterals Make a Parallelogram?

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Date: 2/7/96 at 22:17:46
From: Anonymous

Hello!  I teach grades 7-9 and when we hit quadrilaterals we always
investigate the idea that if you join the midpoints of any quadrilateral
you will always get a parallelogram.  I have asked several teachers
why this is so and have searched through several math books but I
have come up dry.  There are always one or two students who ask me
why this works and I'm tired of replying "I really don't know."  Can
you help me in any way?  Thanks, Susan Wilkie
(Nova Scotia)
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Date: 2/8/96 at 6:11:20
From: Doctor Sarah

Hello there -

You'll find an explanation in our Project of the Month archive at

http://mathforum.org/pom/project2.94.html

formed when the consecutive midpoints of the sides of a quadrilateral
are joined? What if the original quadrilateral were a rectangle? A kite? An
isosceles trapezoid? A square? A rhombus? Other shapes? Explain why you
think your answer is true.   Here's part of the explanation given in the
winning solution sent in by Jennifer Burrows and Amanda Taplett of

"When you connect the consecutive midpoints of a quadrilateral, another
quadrilateral is formed inside of it. We will call the original figure
the mother figure and this new figure the daughter figure. To determine
what the daughter is, you must examine the diagonals of the mother. It
is possible to use the diagonals of the mother because the diagonals are
parallel to the sides of the daughter.

"The type of quadrilateral that is formed can either be a rhombus, a
rectangle, or a square, but it will always be a parallelogram. This is
because when the midpoints are connected to form the sides of the
daughter figure, each side of the mother figure is bisected. Each newly
formed side will be parallel to a diagonal of the mother. Two of the
newly formed sides are parallel to the same diagonal and therefore are
parallel to each other. Along with the other two sides of the daughter
that are parallel to the other diagonal of the mother, a parallelogram
is formed."

There's a lot more explanation, and the solution that received honorable
mention is also worth looking at.

-Doctor Sarah,  The Math Forum

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Associated Topics:
High School Geometry
High School Triangles and Other Polygons
Middle School Geometry
Middle School Triangles and Other Polygons

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