Question 269363
A convex polygon has n sides and 12 n diagonals. Find the value of n .
<pre><font size = 4 color = "indigo"><b>
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:

{{{"nC2"-n=12n}}}

{{{(n(n-1))/2-n=12n}}}

Clear of fractions by multiplying through by 2

{{{n(n-1)-2n=24n}}}

{{{n^2-n-2n=24n}}}

{{{n^2-3n=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</pre>