Question

a polygon has 27 diagonals. how many sides does it have? plz with process

Answer 1

a polygon with 27 diagonals have 9 sides

Answer 2

Given,

No. of diagonals in a polygon = 27


Let, the number of sides is n.

We know that,

⇒ Number of Diagonals in a polygon = [ No. of sides ( No. of sides - 3 ) ] / 2

⇒ 27 = [ n( n - 3 ) ] / 2

⇒ 27 × 2 = n( n - 3 )

⇒ 54 = n² - 3n

⇒ 0 = n² - 3n - 54

⇒ n² - 3n - 54 = 0

Splitting middle term,

⇒ n² - ( 9 - 6 )n - 54 = 0

⇒ n² - 9n + 6n - 54 = 0

⇒ n( n - 9 ) + 6( n - 9 ) = 0

Taking ( n - 9 ) as common,

⇒ ( n - 9 ) ( n + 6 ) = 0

⇒ ( n - 9 ) = 0 ÷ ( n + 6 )

⇒ ( n - 9 ) = 0

•°• n = 9

' Or '

⇒ ( n - 9 ) ( n + 6 ) = 0

⇒ ( n + 6 ) = 0 ÷ ( n - 9 )

⇒ ( n + 6 ) = 0

•°• n = -6

Hence, n = 9 or -6 , but polygon can't have sides as negative.

So, possible value of n is 9.