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A polygon has 54 diagonals. Then, the number of its sides is
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The correct answer is:
12
Number of diagonals in polygon $=54$
Let the number of sides in polygon $=n$
Then,
The number of diagonal in any polygon
$=\frac{n(n-3)}{2}$
$54=\frac{n(n-3)}{2}$
$n^2-3 n=108$
$n^2-3 n-108=0$
$n^2-12 n+9 n-108=0$
$n(n-12)+9(n-12)=0$
$(n+9)(n-12)=0$
$n=-9$ or $n=12$
(not possible)
Hence, the number of sides in polygon is $n=12$
Let the number of sides in polygon $=n$
Then,
The number of diagonal in any polygon
$=\frac{n(n-3)}{2}$
$54=\frac{n(n-3)}{2}$
$n^2-3 n=108$
$n^2-3 n-108=0$
$n^2-12 n+9 n-108=0$
$n(n-12)+9(n-12)=0$
$(n+9)(n-12)=0$
$n=-9$ or $n=12$
(not possible)
Hence, the number of sides in polygon is $n=12$
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