Rhombus and Square ComparisonDate: 01/14/2004 at 22:19:25 From: Meredith Subject: Are All Rhombi Squares Last time my math class got together, a new question arose. If a rhombus can have obtuse and acute angles, how could a square possibly turn into one? One boy thought a square can be a rhombus and a rhombus can be a square. Others only agreed to the first (like me), and still more agreed neither was possible. So it's kind of undecided. Do you have an answer? Date: 01/15/2004 at 09:39:25 From: Doctor Peterson Subject: Re: Are All Rhombi Squares Hi, Meredith. This is a matter of definition. I'll give you an example of a similar kind of definition (simplified): a mammal is a four-legged animal with hair and milk a dog is a mammal that barks So every dog is a mammal, but not every mammal is a dog--some are cats, cows, or whatever. That doesn't mean that a dog is also a cow! Rather, dogs and cows are two different kinds of mammals. Similarly, a rhombus is a quadrilateral whose four sides are all congruent a square is a quadrilateral whose four sides are all congruent and whose angles are all right angles In other words, a square is a rhombus that is also a rectangle These are both rhombuses: +--------+ | | +--------+ | | / / | | / / | | / / +--------+ +--------+ But only this is a square: +--------+ | | | | | | | | +--------+ So every square is a rhombus, because it's a quadrilateral with four congruent (equal) sides. But there are rhombuses that are not square, because their angles are not right angles. The fact that there are rhombuses without right angles doesn't mean that a square has to somehow "turn into" a shape without right angles, any more than a dog has to "turn into" a cow in order to be a mammal. It's just that the definition of a rhombus doesn't mention its angles at all! The angles matter in asking whether it is a square, but for the rhombus only the sides matter. Does that help? - Doctor Peterson, The Math Forum http://mathforum.org/dr.math/ |
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