In a polygon the number of diagonals is 54. The number of sides of the polygon isa)10b)12c)9d)none of theseCorrect answer is option 'B'. Can you explain this answer?

Question

Answer

n(n -3) /2 = 54 ⇒ n = 12

Answer

Understanding the Number of Diagonals in a Polygon
To find the number of sides $ n $ in a polygon with a given number of diagonals, we can use the formula for the number of diagonals:

Formula for Diagonals
The number of diagonals $ D $ in a polygon with $ n $ sides is given by:
$$
D = \frac{n(n-3)}{2}
$$
This formula arises because each vertex connects to $ n-3 $ other vertices (excluding itself and its two adjacent vertices), and since each diagonal is counted twice, we divide by 2.

Setting Up the Equation
Given that the number of diagonals $ D = 54 $, we can set up the equation:
$$
\frac{n(n-3)}{2} = 54
$$
Multiplying both sides by 2 gives:
$$
n(n-3) = 108
$$

Rearranging the Equation
This can be rearranged into a standard quadratic equation:
$$
n^2 - 3n - 108 = 0
$$

Solving the Quadratic Equation
To solve for $ n $, we can use the quadratic formula:
$$
n = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}
$$
Here, $ a = 1 $, $ b = -3 $, and $ c = -108 $.
Calculating the discriminant:
$$
b^2 - 4ac = (-3)^2 - 4(1)(-108) = 9 + 432 = 441
$$
Now, substituting back into the formula:
$$
n = \frac{3 \pm \sqrt{441}}{2}
$$
Since $ \sqrt{441} = 21 $:
$$
n = \frac{3 \pm 21}{2}
$$
This gives two potential solutions:
1. $ n = \frac{24}{2} = 12 $
2. $ n = \frac{-18}{2} = -9 $ (not valid as sides cannot be negative)

Conclusion
Thus, the number of sides of the polygon is $ n = 12 $, confirming that the correct answer is option 'B'.

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