Venn Diagram to Classify QuadrilateralsDate: 01/02/2003 at 14:29:28 From: John Subject: Venn Diagram to classify Quadrilaterals I am looking for a Venn diagram that will accurately display the relation among trapezoids, parallelograms, kites, rhombi, rectangles, and squares. Is a square also a kite? The textbook I am using defines a kite as a quadrilateral having at least two pairs of adjacent sides congruent, no sides used twice in the pairs. Why the "at least two pairs" and the "no sides used twice"? Date: 01/04/2003 at 23:26:44 From: Doctor Peterson Subject: Re: Venn Diagram to classify Quadrillaterals Hi, John. Here's a tree diagram, from which you can make a Venn diagram: Definition of a Trapezoid http://mathforum.org/library/drmath/view/54901.html Quadrilateral / \ / \ Kite Trapezoid | / \ | / \ | Parallelogram Isosceles | / \ Trapezoid | / \ / \ / \ / Rhombus Rectangle \ / \ / \ / Square So let's see if this turns into a nice Venn diagram: +--quadrilateral--------------------------------------+ | | | +--trapezoid------------------------------+ | | | | | | +--parallelogram-----------+ | | | | | | | | | +--------------+--iso trap--+ | | | | | | | | | | +-kite--+-----------+------+ | | | | | | | rhombus | | | | | | | | | | | | | | | | | |square| | | | | | | | | | | | | | | | | | rectangle | | | | | | | +------+-------+------------+ | | | | | | | | | | +-------+------------------+ | | | | | | | | | +--------------------------+ | | | | | | | +-----------------------------------------+ | | | +-----------------------------------------------------+ A label on an edge refers to everything inside; a label inside a region refers to that region only; a label straddling two regions names everything in those regions. So some quadrilaterals are kites, some are trapezoids, and some are neither; some trapezoids are parallelograms, some are isosceles, and some are neither; parallelograms that are also isosceles trapezoids are rectangles; parallelograms that are also kites are rhombuses; those that are both are squares. Note that I am using the inclusive definition of a trapezoid, which not everyone might agree with: a parallelogram is a kind of trapezoid, in which not only one, but two pairs of sides are parallel. Your book's definition of a kite seems awkward. They want to make sure you don't count three consecutive congruent sides as two pairs, so they say you can't use the same side twice; and I can't imagine why they bother saying "at least two pairs," since once you've chosen two disjoint pairs you've used up all the sides. Maybe they want to explicitly allow for the square, in which there are four pairs of congruent sides, giving two ways to choose two that are disjoint to fit the other rule. In any case, the square is a kite. If you have any further questions, feel free to write back. - Doctor Peterson, The Math Forum http://mathforum.org/dr.math/ |
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