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प्रश्न
A convex polygon has 44 diagonals. Find the number of its sides.
उत्तर
Let n be the number of sides in a polygon.
Since, Polygon of n sides has (nC2 – n) number of diagonals
∴ nC2 – n = 44
= `(n!)/(2!(n - 2)!)` – n = 44
= `(n(n - 1)(n - 2)!)/(2*(n - 2)!)` – n = 44
⇒ `(n(n - 1))/2` – n = 44
= `(n^2 - n - 2n)/2` = 44
⇒ n2 – 3n = 44
⇒ n2 – 3n – 88 = 0
= n2 – 11n + 8n – 88 = 0
⇒ n(n – 11) + 8(n – 11) = 0
= (n – 11(n + 8) = 0
∴ n = 11 and n = – 8 ....[∵ n ≠ – 8]
So n = 11
Hence, the required number of sides = 11.
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