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  4. A polygon has 44 diagonals. The number of its sides are

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Q. A polygon has $44$ diagonals. The number of its sides are

AMUAMU 2016Permutations and Combinations

Solution:

Let $n$ be the number of sides of given polygon.
$\therefore $ Polygon with $n$ sides $=\frac{n(n-3)}{2}$ diagonal
$\Rightarrow \frac{n(n-3)}{2}=44 $ [given]
$\Rightarrow n^{2}-3 n=88 $
$\Rightarrow n^{2}-3 n-88=0 $
$\Rightarrow n(n-11)+8(n-11)=0 $
$\Rightarrow (n+8)(n-11)=0$
$\Rightarrow n+8=0 $ or $ n-11=0$
$\Rightarrow n=-8 $ or $ n=11$
But side cannot be negative.
$\therefore n=11$