Quandaries and Queries


Is there ever a case where the diagonals of a trapezoid bisect each other?

Thanks Chris
high school parent




The way to understand this is to play with the possibilities. If you have Geometer's Sketchpad or Cabri, then construct a trapezoid and play with the sketch.

In fact, you will find that it is possible, with an isosceles trapezoid (perhaps an easier sketch to play with). Of course, there are answers to the problem which are not isosceles trapezoids, but you stared wondering about some cases, not necessarily all cases.

As part of the mental play, you could also consider this:
IF the diagonals bisect one another, then the point where they cross is the center of a half turn taking each diagonal onto itself. That is enough to show that everything in the quadrilateral goes onto itself through this half turn.

What are the quadrilaterals with this half turn property? That is an important class you have played with, and all of these are trapezoids. In fact, if you do a half turn, a segment always goes onto a parallel segment - one of the basic properties of parallel lines. So you can start to spin out necessary properties of a bunch of examples here.

The details of how you 'explain' this will depend on which set of 'definitions' you are using for various types of quadrilateral. There is some choice for the 'definition' though your text may not make this clear.

Walter Whiteley