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  4. A polygon has 54 diagonals. Number of sides of this polygon is:

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Q. A polygon has 54 diagonals. Number of sides of this polygon is:

KEAMKEAM 2003

Solution:

$ \frac{n(n-3)}{2}=54 $ $ \Rightarrow $ $ {{n}^{2}}-3n-108=0 $ $ \Rightarrow $ $ {{n}^{2}}-12n+9n-108=0 $ $ \Rightarrow $ $ n(n-12)+9(n-12)=0 $ $ \Rightarrow $ $ (n-12)(n+9)=0 $ $ \Rightarrow $ $ n=12 $ $ (\because n\ne -9) $