Question

A polygon of $ n $ sides has 105 diagonals, then $ n $ is equal to

• A. 20
• B. 21
• C. 15
• D. -14

Hint:

To find the number of diagonals in a polygon, use the formula \( D = \frac{n(n - 3)}{2} \), where \( n \) is the number of sides.

Answer 1

The Correct Option is C

The formula for the number of diagonals \( D \) in a polygon with \( n \) sides is given by: \[ D = \frac{n(n - 3)}{2} \] Substituting \( D = 105 \) into the equation: \[ 105 = \frac{n(n - 3)}{2} \] Multiply both sides by 2: \[ 210 = n(n - 3) \] Solve the quadratic equation: \[ n^2 - 3n - 210 = 0 \] We can solve this using the quadratic formula: \[ n = \frac{-(-3) \pm \sqrt{(-3)^2 - 4(1)(-210)}}{2(1)} \] \[ n = \frac{3 \pm \sqrt{9 + 840}}{2} \] \[ n = \frac{3 \pm \sqrt{849}}{2} \] Approximating the square root: \[ n = \frac{3 \pm 29.14}{2} \] Taking the positive root: \[ n = \frac{3 + 29.14}{2} = \frac{32.14}{2} = 16.07 \] 
Thus, the number of sides is approximately 15. 
Therefore, the correct answer is 15.