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  • Determine the number of sides of a convex polygon with 135 diagonals.Option: 1 12Option: 2</s

Determine the number of sides of a convex polygon with 135 diagonals.

Option: 1

12


Option: 2

15


Option: 3

18


Option: 4

21


Answers (1)

best_answer

In this case, let's say that the polygon has n sides. Therefore, there will also be 'n' vertices.

If we connect any of these n points to form a line, the result will either be a diagonal or a side of the polygon.

Thus, there are ^{n}C_{2} methods in total to choose two points from a total of n points.

The total number of lines formed using n vertices = Number of diagonal + Number of sides.

\mathrm{\begin{aligned} &{ }^n C_2=135+n\\ &{ }^n C_2-135=n\\ &\frac{n !}{2 !(n-2) !}-135=n \end{aligned}}

We can write \mathrm{\mathrm{n!} as n !=n(n-1)(n-2)!}
\mathrm{ \begin{aligned} & \frac{n(n-1)(n-2) !}{2 !(n-2) !}-n=135 \\ & \frac{n(n-1)}{2}-n=135 \\ & \frac{n^2-3 n}{2}=135 \end{aligned} }
Rearranging the equation to get,

\mathrm{ n^2-3 n-270=0 }

Solving the quadratic equation to get,

\mathrm{ n=18 \text { and } n=-15 }

The value of n cannot be negative.

Therefore, the number of sides in a polygon is 18.

 

 

Posted by

Suraj Bhandari

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