Question

How many sides will be there in a polygon having 54 diaggonals?

• A. 27
• B. 108
• C. 54
• D. 12

Answer 1

The Correct Option is D

The formula for the number of diagonals \( D \) in a polygon with \( n \) sides is given by: \[ D = \frac{n(n - 3)}{2} \] We are given that the number of diagonals is 54. So, we set up the equation: \[ 54 = \frac{n(n - 3)}{2} \] Multiplying both sides by 2: \[ 108 = n(n - 3) \] Expanding the equation: \[ 108 = n^2 - 3n \] Rearranging the equation: \[ n^2 - 3n - 108 = 0 \] Solving this quadratic equation using the quadratic formula: \[ n = \frac{-(-3) \pm \sqrt{(-3)^2 - 4(1)(-108)}}{2(1)} \] \[ n = \frac{3 \pm \sqrt{9 + 432}}{2} \] \[ n = \frac{3 \pm \sqrt{441}}{2} \] \[ n = \frac{3 \pm 21}{2} \] Thus, we get two solutions: \[ n = \frac{3 + 21}{2} = 12 \quad \text{or} \quad n = \frac{3 - 21}{2} = -9 \] Since the number of sides \( n \) must be positive, we choose \( n = 12 \).

The correct option is (D): \(12\)