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we know that we can triangulate any convex polygonal region .can we parallelogramulate a convex region bounded by a 2009-gon????????????????????? plzzzzzz explain how to solve.................

we know that we can triangulate any convex polygonal region .can we parallelogramulate a convex region bounded by a 2009-gon?????????????????????        plzzzzzz    explain how to solve.................

Grade:Upto college level

1 Answers

OMG
12 Points
6 years ago
It cannot be done. Heres the reason why.
Assume a n-gon with vertices labelled 1,2,3,4,5,...n (in that order).
To parralelogramulate a polygon we need to create a quadrilateral using just lines and not adding any extra points within or outside the region.
So the first quadrilateral with be created using vertices 1-2-3-4. where a new line is made by joining 1 and 4. So the polygon now will be reduced to 1-4-5-6-7-8...-(n-1)-n. The polygon has been reduced by 2 vertices. We continue this process till we have a polygon of size 4 or 5.
If its 5 then we cannot parallelogramulate a convex region bounded by a n-gon.
If its 4 then we can.
Now it will be 5 only when n is odd. So 2009-gon cannot be parallelogramulated.

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