Date: 12/01/2003 at 16:44:51 From: Erik Subject: Is it possible to form a polygon with two straight lines? My 5th grade math teacher said that we had to draw a polygon using two straight lines. We are not allowed to use curved lines. I think this is not possible. I thought that a polygon had to have at least 3 sides. It had to have line segments that meet at two sides at each vertex. Am I wrong?
Date: 12/01/2003 at 18:53:39 From: Doctor Ian Subject: Re: Is it possible to form a polygon with two straight lines? Hi Erik, It depends on how flexible you're willing to be in your definition of 'straight' and 'polygon'. For example, one definition of a 'straight' segment is that it connects its endpoints with the shortest possible path. Find a globe, and look at one of the meridians of longitude (which run from pole to pole). Pick any two points on that meridian. Now, the shortest path ALONG THE SURFACE of the globe is that meridian. So while in three dimensions we think of it as a curve, ON THE SURFACE of the globe, it's a straight line. Now, pick any two meridians--say 0 degrees and 30 degrees. They intersect at the north and south poles, and enclose an area between them, right? So now you have two straight line segements, which intersect at vertices, and enclose an area. That looks like a polygon to me! Or, you could try this. Take a lined piece of paper, and draw two intersecting lines across is, so that they leave the paper at the same heights on each side: +---------------+ | | A A' | . . | | . . | | . | | . . | | . . | B B' | | +---------------+ Now roll the paper into a cylinder, so that point A touches point A', and point B touches point B'. Once again, you have straight line segments meeting to form vertices, and enclosing an area. However, if you restrict yourself to using flat spaces (like a piece of paper that you can't pick up), then you're correct: two lines can form only one angle, so they can't enclose a polygon. What curving the space allows you to do is have the lines intersect more than once. Unless, of course, you're willing to draw a 'degenerate' polygon: http://mathworld.wolfram.com/Degenerate.html In that case, you could just draw a line segment from point A to point B, and another one from point B to point A. This is the limiting case of a rectangle, as the lengths of one pair of opposite sides goes to zero. But not everyone would be willing to call it a polygon. Does this help? - Doctor Ian, The Math Forum http://mathforum.org/dr.math/
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