Cutting the plane

Geometry Level 3

A polygon has 2017 number of sides. If each point of a diagonal is inside of the polygon, then we color the diagonal red.

What is the minimum value of the number of red diagonals?

2015 None of the others 2016 2017 2014 2018 2013

1 solution

Zach Abueg
Jul 31, 2017

For any $n$ -gon, there are $n - 3$ internal diagonals. This is simply because there are $n$ vertices for diagonals to come from, and $3$ they cannot go to: themselves, and the $2$ vertices directly adjacent to them, since the segments connecting those vertices are actually sides.

Thus, for $n = 2017$ , we have $\boxed{2014}$ red diagonals.

Hm, can you elaborate? I agree that from each vertex, there are $n - 3$ diagonals because we have to exclude 3 vertices. So there are a total of $\frac{ n(n-3) } { 2}$ diagonals.

How does that imply that in an n-gon, there are at least $n-3$ internal diagonals?

Calvin Lin Staff - 3 years, 10 months ago

Cutting the plane

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