[DivineNathicana]

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.

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I have the most trouble with the "show all steps used to obtain the answer" part. lol

Anyone have any ideas?

- Alisa

[maddog]

It is called a spline. The method is "Least Squares Analysis".

Maddog

[Bo]

does the least square method equally devide the points in two? in general i don't think so... (altough i can't prove it ) (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 )

Bo

[DivineNathicana]

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.

[DivineNathicana]

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? = /

[DivineNathicana]

Originally Posted by Bo

- 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?

[DivineNathicana]

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. "

[Fishteacher73]

Are these 1000 random points? Is there a patern to the distribution?

[DivineNathicana]

No pattern whatsoever. Just random points. And this is supposed to be possible with ANY arrangement of them, too. = (

[DivineNathicana]

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?...icp=1&.intl=us

Thank you!