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Each vertex of a nine-sided polygon is connected to the other vertices with line segments.  How many of these line segments lie inside the polygon?

 Jun 17, 2020
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Here are two solutions.

 

1) It takes 2 vertices to make a line, so there are \({9 \choose 2}\) line segments. We have to subtract 9 from this because the 9 sides don't count.

\({9 \choose 2}-9=27\)

 

2) From each vertice, there are 6 other vertices to draw a diagonal from (we can't include the vertice itself or its two adjacent vertices). In all, there are \(\frac{9*6}{2}=27\). We divide by 2 because we count each diagonal twice (once from each endpoint). 

 Jun 17, 2020

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