Question

A Polygon has 44 diagonals, then the numbers of its sides are:
1. 11
2. 7
3. 8
4. None of these

Answer 1

Correct Answer - Option 1 : 11

Concept:

• If the number of sides of this polygon is n, therefore the number of vertices will also be n.
• If we draw a line by joining any of these n points, the lines will either be diagonal or a side of the polygon.
• Hence, the total number of lines that can be drawn using these points is equal to the total number of ways of selecting two points out of n points.
• A total number of lines formed using n vertices = Number of diagonal + Number of sides.

 

Formula used:

The total number of ways of selecting r objects out of n objects ⁿC_r

Where, \(^nC_r\ =\ \frac{n!}{r!(n\ -\ r)!}\)

Number of diagonal which can be drawn from n side polygon =  nCr - n

Calculation:

Given that, 

Number of diagonal  = 44

Let, number of sides is n

We know that,

Number of diagonal from n side = nCr - n

\(⇒\ 44 = {^nC_2}-n = \frac{n!}{r!(n\ -\ r)!}\)

\(⇒\ 44 = \frac{1}{2}n(n-1)-n\)

⇒ n² - 3n - 88 = 0

⇒ n2 - 11n + 8n - 88 = 0

⇒ n(n - 11) + 8(n - 11) = 0

⇒  n = 11 and n = - 8

since n ≠ - 8

Hence, total number of side is 11.