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Lines dividing a plane into regions problem

Expert replies
by winnerhere » Mon Jul 19, 2010 6:23 pm
Which of the following is NOT a possible number of regions into which three stright lines(of infinite extent) can divide a plane?

1) 7

2) 6

3) 5

4) 4

5) None of these


Thanks,
Sai
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Source: — Problem Solving |

by sk818020 » Mon Jul 19, 2010 6:36 pm
I'd say 5 sections or answer 3. 4 and 6 should be easy enough. 4 sections being possible when 2 lines intersect with the other line overlapping one of the others. 6 sections being possible when all 3 lines converge at a single point, while not being parallel. 7 sections would be possible when the three lines make a triangle, six outter section with the inner triangle section being the seventh. I cannot conceive a way to make 5 though.

Thus, 3 is the answer.

Can you please let me know what the OA is?

Thanks,

Jared[spoiler][/spoiler]
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by winnerhere » Mon Jul 19, 2010 7:07 pm
you are right. But what about

2 lines intersection and 1 line not intersecting any of those lines....


Image

please refer the pic. considering the square as the plane..these lines create 5 regions..right?

what am i missing in logic..please throw some light on this :)

thanks for quick response :)
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by sk818020 » Mon Jul 19, 2010 7:12 pm
winnerhere wrote:you are right. But what about

2 lines intersection and 1 line not intersecting any of those lines....


Image

please refer the pic. considering the square as the plane..these lines create 5 regions..right?

what am i missing in logic..please throw some light on this :)

thanks for quick response :)
Ahh, but your rectangle would count as 4 more lines enclosing an otherwise infinitesimal plane. The lines are infinite, thus the plane must be infinite, for a finite plane cannot contain infinite lines. The lines are infinite. This is a given in the question. They are not cut off by some arbitrary box you put them in. For any line to be infinite it must exist on at least an infinite plane. So, your attached example, assuming we rid ourselves of the rectangular parameter, would be no more than an example of the triangle I described above as the top-left line would extend into infinity and create one.

Thanks,

Jared
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by sk818020 » Mon Jul 19, 2010 7:21 pm
winnerhere wrote:you are right. But what about

2 lines intersection and 1 line not intersecting any of those lines....


Image

please refer the pic. considering the square as the plane..these lines create 5 regions..right?

what am i missing in logic..please throw some light on this :)

thanks for quick response :)
One other note. It is impossible for infinite lines to not intersect unless those infinite lines overlap each other (or mathematically speaking they have the same slope, x and y intercept). So, again, to stress the infinite part of this, these lines are infinite and continue into eternity. If each line does not have the exact same linear formula, then they will intersect.

Jared
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by winnerhere » Mon Jul 26, 2010 7:00 pm
ohh..got it!

thanks dude :)
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