Maximum Number of Intersections of n Distinct Lines
Date: 10/07/98 at 23:41:03 From: Sally Subject: Algebra 2 Find a pattern for the maximum number of intersections of n lines, where n is greater than or equal to 2. I've tried drawing pictures but in geometry class I was taught that the number of intersections of a line is infinite, because the line never ends. Was I informed wrong when I was in geometry? I'm really confused with this whole problem. Please help. Thanks a lot.
Date: 10/08/98 at 12:55:21 From: Doctor Rick Subject: Re: Algebra 2 Hi, Sally. I'm not exactly sure what you are thinking of when you say the number of intersections of a line is infinite, but in at least one sense, you are right, and the problem needs to be restated. Let's just look at two lines. There are three possibilities according to Euclidean geometry: 1. The lines are parallel, and they do not intersect. 2. The lines intersect in exactly one point. 3. The lines are the same line, so they intersect at every point along the line. (These possibilities arise from Euclid's Parallel Postulate; other kinds of geometry have been invented in which this is not true. For instance, two "straight lines" on the surface of the earth - or a sphere - will meet in exactly two points, on opposite sides of the earth.) It's that third case that causes trouble. Is that what you are thinking about? The problem should really have said, "Find a pattern for the maximum number of intersections of n _distinct_ lines, where n is greater than or equal to 2." Then the maximum number is not infinite, and it's much more interesting. To get you started, think about a triangle - extend its line segments into lines, and you have 3 lines intersecting in 3 points. If you do the same with a square, you get two pairs of parallel lines; you can get more intersections if they are not parallel. See what you can do. Have fun with it! - Doctor Rick, The Math Forum http://mathforum.org/dr.math/
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