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Suppose $n$ straight lines are drawn on a plane. When these lines are removed, the plane falls apart into several connected components called regions. $A$ region $R$ is said to be convex if it has the following property: whenever two points are in $R$, then the entire line segment joining them is in $R$. Suppose no two of the n lines are parallel. Which of the following is true?

  1. $O (n)$ regions are produced, and each region is convex.
  2. $O (n^{2})$ regions are produced but they need not all be convex.
  3. $O (n^{2})$ regions are produced, and each region is convex.
  4. $O (n \log n)$ regions are produced, but they need not all be convex.
  5. All regions are convex but there may be exponentially many of them.
in Quantitative Aptitude
edited by
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1 Answer

1 vote
Option C must be correct.

Regions divided by n lines = 1+1+2+3+4+..........+n-1 = 1 + n(n-1)/2 = O(n^2)

option B can be easily discarded as convex segments are obvious.

Hence C is correct.

edited by
0
how non-convex region is forming?
0

I think answer should be C..

0
sorry, my sloppy mistake.
all regions are convex.

Option C is correct.

(edited the answer)
0

No-convex regions are possible, isn't it?

Please correct me if i am wrong.

 

1

@Nitesh Tripathi I think region divided by n lines should be ---. n ( n + 1 ) / 2 + 1

See this https://www.cut-the-knot.org/proofs/LinesDividePlane.shtml

Answer:

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