What are some irrefutable arguments that a square is a rectangle? My
family thinks a rectangle must be oblong, and believes I am nuts for
saying otherwise. I need to convince them.

noseredla-ga,


You are right, of course.

But how to convince the non-believers?

A good starting place might be the wonderful internet resource known
as Wikipedia -- an online encyclopedia.  The Wikipedia definition of a
rectangle can be found here:


http://en.wikipedia.org/wiki/Rectangle


---
Rectangle:  In geometry, a rectangle is defined as a quadrilateral
polygon in which all four angles are right angles...From this
definition, it follows that a rectangle has two pairs of opposite
sides of equal length; that is, a rectangle is a parallelogram.
---



This is enough to convince me (and you, I take it) that a square is
indeed a rectangle.  But it may not be enough for your family members.

Happily, the Wikipedia definition continues with this very
straightforward statement:


---
A square is a special kind of rectangle where all four sides have
equal length; that is, a square is both a rectangle and a rhombus.
---


Not enough?  Then how about the Wikipedia definition of s Square:


http://en.wikipedia.org/wiki/Square_%28geometry%29



which states as clearly as can be:


---
...Squares are regular quadrilaterals, rectangles, rhombi, kites,
parallelograms, and isosceles trapezoids/isosceles trapezia.
---



Need more?  How about this "Ask Dr Math" entry:


http://mathforum.org/library/drmath/view/65096.html
Are Squares Rectangles? Are Rectangles Squares?



The answer from Dr. Math is, of course, "Actually, EVERY square is a
rectangle, since the angles in a square are always right angles. 
That's more than saying that a square CAN be a rectangle; it IS."



I trust these links will meet your needs and help convince the skeptics among you.


However, if you find you need additional information, just let me know
by posting a "Request for Clarification" and I'll be happy to drum up
some additional support.


Yours in geometry,

pafalafa-ga


search strategy -- Used bookmarked sites for Wikipedia, and search
Google for [ square rectangle ]

---

Request for Answer Clarification by nosredla-ga on 05 Oct 2005 14:43 PDT

Excellent answer - clear as could be!

What are rhombi, kites, and isosceles trapezoids/isosceles trapezia?
rhombi, kites?

---

Clarification of Answer by pafalafa-ga on 05 Oct 2005 16:00 PDT

Glad to help, though let me just point out that each of the underlined
terms in the Wikipedia definitions (those that are in blue type) are
themselves Wikipedia entries.

Clicking on 'kites' for instance, takes you to:


http://en.wikipedia.org/wiki/Geometric_kite


where you learn that a geometric kite is a a four-sided polygon with
two pairs of equal sides, and the equal sides are adjacent.  If it's
hard to visualize, think of a ....er....kite!


Similarly, the other terms are linked as well, and if you click on
them you can see that:


rhombi is the awkward plural of rhombus, sort of a skewed square that
typically looks like a diamond.

An isosceles trapezoid is another 4-sided figure, with two sides
parallel and two that are not parallel -- the non-parallel sides are
equal in length.  "Trapezia" is just another way of saying trapezoids.


Hope that does the trick.


paf

Answer was quick, complete, cited multiple sources, and displayed a
nice sense of humour. I'm very impressed.