Classifying QuadrilateralsDate: 01/25/2006 at 09:11:07 From: adrienne Subject: Why is a square a rectangle My teacher says that a square is also a rectangle. I don't understand how that can be since the sides are the same length. Date: 01/25/2006 at 10:12:45 From: Doctor Peterson Subject: Re: Why is a square a rectangle Hi, Adrienne. In everyday usage, we would point to a square and say "that's a square, not a rectangle." That's because we generally name everything as specifically as possible. "That's not a lady, that's my wife!" In math and science, we have a slightly different perspective on words: we want each term to apply to anything for which it makes sense, and we want each definition to be as straightforward as possible. So, for example, we define a rectangle simply as "a figure with four straight sides, all of whose angles are right angles". If you compare a square with this definition, you see that it fits: it does have four sides, and it does have four right angles. So it is a rectangle. It's MORE than a rectangle, of course; in addition to the requirements of a rectangle, it also has four EQUAL sides. But that doesn't make it NOT a rectangle, only a more SPECIAL rectangle. We make definitions like this (called "inclusive definitions") for a reason. When we state a theorem about rectangles, we want to be able to apply it in every case where it is true. Since any fact about a shape that depends only on the fact that it has right angles will apply not only to "mere rectangles", but also to squares (special rectangles), it makes sense to use one word to cover them all, rather than having one word for "non-square rectangles", and another for squares. If nothing else, this makes it a lot easier to state theorems. The same is true in other fields of science. We use inclusive definitions in naming animals, for example: a terrier is a special kind of dog, and a dog is a special kind of mammal, and a mammal is a special kind of vertebrate. You wouldn't say "that's a terrier, not a mammal" just because "terrier" is a more specific term than "mammal"; we need to have a word that covers all mammals, so that we can talk about facts that are true of all of them. In the same way, a square is just a "species" of rectangle. Just as we can make a whole classification tree for animals, we can classify shapes using these inclusive names. Here is a classification of the main types of quadrilaterals (four-sided figures): quadrilateral / \ / \ / \ kite trapezoid | / \ | / \ | / \ | parallelogram isosceles | / \ trapezoid | / \ / | / \ / rhombus rectangle \ / \ / \ / square Each figure is a special case of the figure(s) above it. Without inclusive naming, we would have to write theorems like "if you connect the midpoints of successive sides of a quadrilateral or kite or trapezoid or parallelogram or rhombus or rectangle or square, then the resulting figure will be a parallelogram or rhombus or rectangle or square." Using inclusive definitions, the theorem is just "if you connect the midpoints of successive sides of a quadrilateral, then the resulting figure will be a parallelogram." That saves a lot of trees! The following page talks about both the general issue of inclusive definitions, and the specific issue of naming quadrilaterals, including squares and rectangles: Quadrilateral Classification http://mathforum.org/library/drmath/sets/select/dm_quad_classify.html 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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