SOLUTION: A convex polygon has n sides and 12 n diagonals. Find the value of n .

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Question 269363: A convex polygon has n sides and 12 n diagonals. Find the value of n .
Answer by AnlytcPhil(1806) About Me  (Show Source):
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A convex polygon has n sides and 12 n diagonals. Find the value of n .

There are nC2 ways to choose two vertices to join.  Every diagonal
is formed by joining two vertices. The only ways to join two vertices 
which will not determine a diagonal is to join two adjacent vertices,
which will form a side, not a diagonal.  So we only need to subtract 
the n sides from the nC2 ways to join two vertices and we get 

nC2+-+n

diagonals.  Since there are 12n diagonals we have this equation:

%22nC2%22-n=12n

%28n%28n-1%29%29%2F2-n=12n

Clear of fractions by multiplying through by 2

n%28n-1%29-2n=24n

n%5E2-n-2n=24n

n%5E2-3n=24n

Divide through by n, which is OK since n can't be 0.

n-3=24

n=27

The polygon has 27 sides.

Edwin