Question

The Difference between any two consecutive interior angles of a polygon is 5° If smallest angle is 120° find the no. if sudes

Answer 1

Hey !
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Ques. → The Difference between any two consecutive interior angles of a Polygon is 5°. If the smallest angle is 120° , find the number of its sides .


Ans. → The angles of the polygon will from an Arithmetic Progression ( A. P ) with the first term a = 120 and common difference d = 5


•°• We have the A.P : - 120 , 125 , 130...........


=>Also , we know that the sum of the sides ( n) for a polygon = ( n - 2 ) × 180


★Using the formula for Sn and equating it with the formula for n sides we have ,


=> n/2 [ 2a + (n-1)d ] = 180 × ( n -2 )


=> n/2 [ 2 × 120 + ( n - 1 ) 5 ] = 180 ( n -2)


=> Solving it we have the following Quadratic equation →


» n² - 25n + 144 = 0

[ Using Middle Term splitting ]


=> n² - 16n - 9n + 144 = 0


=> n ( n - 16 ) - 9 ( n - 16)

=> ( n - 16 ) ( n - 9)


•°• n = 16 , 9

→ This means the polygon can have either 16 or 9 sides. Both values are possible !


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Answer 2

120 , 125 , 130...........is an AP



n/2 [ 2a + (n-1)d ] = 180 × ( n -2 )


n/2 [ 2 × 120 + ( n - 1 ) 5 ] = 180 ( n -2)




= n² - 25n + 144 = 0




=n² - 16n - 9n + 144 = 0


= n ( n - 16 ) - 9 ( n - 16)

=> ( n - 16 ) ( n - 9)


it has r 16 or 9 sides




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