Why do the Midpoints of Quadrilaterals Make a Parallelogram?
Date: 2/7/96 at 22:17:46 From: Anonymous Subject: Geometry-midpoints of quadrilaterals 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)
Date: 2/8/96 at 6:11:20 From: Doctor Sarah Subject: Re: Geometry-midpoints of quadrilaterals Hello there - You'll find an explanation in our Project of the Month archive at http://mathforum.org/pom/project2.94.html The question was about Midpoints and Quadrilaterals: What figure is 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 Greenwich Academy, Greenwich, CT: "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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