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
.
~ cxsmi
See Also
1950 AHSC (Problems • Answer Key • Resources) | ||
Preceded by Problem 44 |
Followed by Problem 46 | |
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 • 16 • 17 • 18 • 19 • 20 • 21 • 22 • 23 • 24 • 25 • 26 • 27 • 28 • 29 • 30 • 31 • 32 • 33 • 34 • 35 • 36 • 37 • 38 • 39 • 40 • 41 • 42 • 43 • 44 • 45 • 46 • 47 • 48 • 49 • 50 | ||
All AHSME Problems and Solutions |
The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions.