A polygon has 54 diagonals, then the number of its sides arc :a)11b)12c)7d)noneCorrect answer is option 'B'. Can you explain this answer?

Question

Answer

To determine the number of sides in a polygon given the number of diagonals, we can use a mathematical formula.

Understanding the Formula for Diagonals
The formula to calculate the number of diagonals (D) in a polygon with 'n' sides is:
D = n(n - 3) / 2
Where:
- D = number of diagonals
- n = number of sides

Given Information
In this case, we know that D = 54. Therefore, we can set up the equation:
54 = n(n - 3) / 2

Solving the Equation
To eliminate the fraction, multiply both sides by 2:
108 = n(n - 3)
Now, expand and rearrange the equation:
n² - 3n - 108 = 0
This is a quadratic equation in standard form.

Applying the Quadratic Formula
We can solve this equation using the quadratic formula, which is:
n = [ -b ± √(b² - 4ac) ] / 2a
Here, a = 1, b = -3, and c = -108. Plugging in these values:
n = [ 3 ± √((-3)² - 4 * 1 * (-108)) ] / 2 * 1
Calculate the discriminant:
= [ 3 ± √(9 + 432) ] / 2
= [ 3 ± √441 ] / 2
= [ 3 ± 21 ] / 2

Finding the Values of n
Calculating the two possible values:
1. n = (3 + 21) / 2 = 12
2. n = (3 - 21) / 2 = -9 (not valid since sides cannot be negative)
Thus, the number of sides in the polygon is:

n = 12
Hence, the correct answer is option 'B'.

Answer

We have a very good formula

where x in number of diagonal and n stand for number of sides so putting the value.

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