the interior angles of a polygon are in AP the smallest angle is 120° and the common difference is 5° . find the number of sides of the polygon.
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The right answer is 9
KingAryan:
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a=120 d=5
Sum of angles in a polygon=180×(n-2) -- (i)
n= no. of sides of the polygon
Sum of all angles in Ap=
Sn = n/2 (2a+(n-1)d) --(ii)
(i)=(ii)
180×(n-2)=n/2 (2a+(n-1)d)
180×2 × (n-2) = 2an + dn^2 - dn
360n - 720 = 2×120×n + 5×n^2 -5×n
360n-720= 235n+5n^2
5n^2-125n+720=0 --- (÷5)
n^2-25+144=0
Using quadratic forfumala
x= -b+or- root D / 2a
x= 25 + 7 / 2 = 16
Or
x= 25-7 / 2 = 9
therefore no of sides 9 or 16
Sum of angles in a polygon=180×(n-2) -- (i)
n= no. of sides of the polygon
Sum of all angles in Ap=
Sn = n/2 (2a+(n-1)d) --(ii)
(i)=(ii)
180×(n-2)=n/2 (2a+(n-1)d)
180×2 × (n-2) = 2an + dn^2 - dn
360n - 720 = 2×120×n + 5×n^2 -5×n
360n-720= 235n+5n^2
5n^2-125n+720=0 --- (÷5)
n^2-25+144=0
Using quadratic forfumala
x= -b+or- root D / 2a
x= 25 + 7 / 2 = 16
Or
x= 25-7 / 2 = 9
therefore no of sides 9 or 16
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