1950 AHSME Problems/Problem 45
Problem
The number of diagonals that can be drawn in a polygon of 100 sides is:
Solution
Each diagonal has its two endpoints as vertices of the 100-gon. Each pair of vertices determines exactly one diagonal. Therefore the answer should be . However this also counts the 100 sides of the polygon, so the actual answer is .
Solution 2
We can choose vertices for each vertex to draw the diagonal, as we cannot connect a vertex to itself or any of its two adjacent vertices. Thus, there are diagonals, because we are overcounting by a factor of (we are counting each diagonal twice - one for each endpoint). So, our answer is .
Solution 3
The formula for the number of diagonals of a polygon with sides is . Taking , we see that the number of diagonals that may be drawn in this polygon is or .
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See Also
1950 AHSC (Problems • Answer Key • Resources) | ||
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Followed by Problem 46 | |
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