DivineNathicana Posted March 1, 2005 Report Share Posted March 1, 2005 A thousand points are graphed in the coordiante plane. Explain why it is possible to draw a straight line in the plane so that half of the points are on one side of the line and half are on the other. (Hint: Consider the slopes of the lines determined by each pair of points.) N.B. You must prove this in general, not just for one specific case. I.e.: describe the process someone would take that given any thousand points, he or she could draw a line that divides them into two halves. Show all steps used to obtain the answer. --------------------------------------------------------------------------------------------------------- I have the most trouble with the "show all steps used to obtain the answer" part. lol Anyone have any ideas? - Alisa Link to comment Share on other sites More sharing options...
maddog Posted March 1, 2005 Report Share Posted March 1, 2005 It is called a spline. The method is "Least Squares Analysis". Maddog Link to comment Share on other sites More sharing options...
Bo Posted March 1, 2005 Report Share Posted March 1, 2005 does the least square method equally devide the points in two? in general i don't think so... (altough i can't prove it :eek:) (my feeling is that, since the LSM is about the deviation of the line you draw, it doesn't say anything about the number of points) what i think is the correct way is this:- devide the set with a vertical line exactly in 2.- Compute for each set the mean point (<x>,<y>)(that is: <x> = (x1+x2+x3....+xn)/n), idem for y)- the straight line between the 2 mean points divides the points exactly in 2. i'm quite sure this is the right way; a formal proof of why this works is tricky... (maybe later more :eek:) Bo Link to comment Share on other sites More sharing options...
DivineNathicana Posted March 1, 2005 Author Report Share Posted March 1, 2005 Yeah, I don't think that Least Squares Analysis will work for this because it would give you the mean of the coordinates of the points, and a different number can be above the mean than the number below the mean. Link to comment Share on other sites More sharing options...
DivineNathicana Posted March 1, 2005 Author Report Share Posted March 1, 2005 By the way, the hint was to consider the slopes of the lines determined by each pair of points. What can we do with that? = / Link to comment Share on other sites More sharing options...
DivineNathicana Posted March 1, 2005 Author Report Share Posted March 1, 2005 - devide the set with a vertical line exactly in 2. Bo, by that did you mean that we arbitrarily divide the points into two parts, or do we divide them by counting out 500 points and drawing a line between the two halves? Link to comment Share on other sites More sharing options...
DivineNathicana Posted March 1, 2005 Author Report Share Posted March 1, 2005 You also have to explain the WHY: "Explain why it is possible to draw a straight line in the plane so that half of the points are on one side of the line and half are on the other. " Link to comment Share on other sites More sharing options...
Fishteacher73 Posted March 2, 2005 Report Share Posted March 2, 2005 Are these 1000 random points? Is there a patern to the distribution? Link to comment Share on other sites More sharing options...
DivineNathicana Posted March 2, 2005 Author Report Share Posted March 2, 2005 No pattern whatsoever. Just random points. And this is supposed to be possible with ANY arrangement of them, too. = ( Link to comment Share on other sites More sharing options...
DivineNathicana Posted March 3, 2005 Author Report Share Posted March 3, 2005 Hey guys, I found a solution to the problem but I don't exactly understand it because of my diminutive mathematical education (I didn't start pre-cal yet). Can someone please explain it to me? It's #4, all the way at the bottom. http://216.109.117.135/search/cache?p=%22half+of+the+points+are+on+one+side%22&ei=UTF-8&fl=0&u=www.cs.rutgers.edu/%7Efarach/test2sol/test2sol.html&w=%22half+of+the+points+are+on+one+side%22&d=FF423545EC&icp=1&.intl=us Thank you! Link to comment Share on other sites More sharing options...
Tim_Lou Posted March 3, 2005 Report Share Posted March 3, 2005 suppose we line up the dots from left to right accordingly, when 2+ points share the same x coordiante, we go from top to bottom.a continuous "line" made up of line segments is formed.now we have to prove that a line can be drawn in the middle line segment that does not cross any line segments other than the middle one.since we draw the "line" from left to right, top and bottom, no enclosed area is form, therefore, the dividing line doesnt necessary have to cross other line segments. i dont know if its good proof... hope it helps. Link to comment Share on other sites More sharing options...
Bo Posted March 4, 2005 Report Share Posted March 4, 2005 hmm i don't understand the site you posted either (altough it does appear to be a sollution :hyper:) (the site is written by computer people, not by mathematicians i think....)anyway i am quite sure the following idea should be your basis:with 4 points: p1, p2, p3, p4:take the mean between p1 and p2, (m1) and take the mean between p3 and p4. (m2). The line through m1 and m2 splits the set in 2. Bo Link to comment Share on other sites More sharing options...
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