A polygon having 27 diagonals , how many sides does it have?
Answers
Answer:
Hey mate here is the answer
Answer is 9 sides.
Hope it helps you
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Answer:
Step-by-step explanation:
→ The formula for the no. of diagonals of a ’n' sided polygon is n(n-3)/2 .
It works for triangle (0 diagonals) , quadrilateral (2 diagonals) , pentagon (5 diagonals) and for every another polygon.
Let's come to the problem.
No. Of diagonals = n(n-3)/2 = 27
n(n-3) = 54
n^2 - 3n -54 = 0
n^2 - 9n + 6n - 54 = 0
n(n-9) + 6(n-9) = 0
(n-9)(n+6) = 0
n = 9, -6
(excluding -6)
n = 9
The polygon has 9 sides.
It is a Nonagon.
Hurry! We found it.
The formula I mentioned can be derived.
No. of sides = No. of vertices = n
Each diagonal joins two of the vertices.
The no. of ways (n) vertices can be joined (two at a time) is nC2.
But we are also including (n) sides by joining two vertices at a time.
To get only the no. of diagonals, we must subtract the (n) sides from nC2.
Hence,
No. of diagonals = nC2 - n = (n(n-1)/2!) - n
= (n(n-1) - 2n)/2 = n(n-3)/2
(OR)
→ We know that, to find diagonals of a polygon, we use the formula n(n-3)/2 where n is the number of sides. It is given that the polygon have 27 diagonals.
So,
=>n(n-3)/2=27
=>n²-3n=27×2
=>n²-3n=54
=>n²-3n-54=0
=>n²-(9–6)n-54=0
=>n²-9n+6n-54=0
=>n(n-9)+6(n-9)=0
(n+6)=0; (n-9)=0
n=-6; n=9
n should not be negative so 9 is the number of sides.
So the polygon have 9 sides which is also called a nanogon.
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