Question

If the number of diagonals of a regular polygon of \( n \) sides is 104, then \( n = \)

• A. 19
• B. 16
• C. 13
• D. 11

Hint:

Use the formula for the number of diagonals of a polygon to solve for \( n \), then solve the resulting quadratic equation.

Answer 1

The Correct Option is B

The formula for the number of diagonals \( D \) of a regular polygon with \( n \) sides is: \[ D = \frac{n(n - 3)}{2} \] Given \( D = 104 \), we substitute into the formula: \[ \frac{n(n - 3)}{2} = 104 \] \[ n(n - 3) = 208 \] \[ n^2 - 3n - 208 = 0 \] Solving this quadratic equation: \[ n = \frac{-(-3) \pm \sqrt{(-3)^2 - 4(1)(-208)}}{2(1)} = \frac{3 \pm \sqrt{9 + 832}}{2} = \frac{3 \pm \sqrt{841}}{2} \] \[ n = \frac{3 \pm 29}{2} \] Thus, \( n = \frac{3 + 29}{2} = 16 \) or \( n = \frac{3 - 29}{2} = -13 \). Since \( n \) must be a positive integer, we have \( n = 16 \).