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

Question

Smart Achievers · Accepted Answer

(1)

$Polygon has 54 diagonals.$

$Number of diagonals in a polygon of n sides = \frac{n (n - 3)}{2}$

$Here, \frac{n (n - 3)}{2} = 54$

$\Rightarrow n (n - 3) = 108$ $\Rightarrow n^{2} - 3 n - 108 = 0$

$\Rightarrow n^{2} - 12 n + 9 n - 108 = 0$

$\Rightarrow n (n - 12) + 9 (n - 12) = 0$ $\Rightarrow (n + 9) (n - 12) = 0$

$∴$ $n = 12$ $[∵ n = - 9 is not possible]$

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