Questions: The equation D=1/2 n(n-3) gives the number of diagonals D for a polygon with n sides. Use this equation to find the number of sides n for a polygon that has 77 diagonals. The polygon has sides.

Solution Steps

The equation for the number of diagonals \( D \) in a polygon with \( n \) sides is given by:

\[ D = \frac{1}{2} n(n - 3) \]

We are given that \( D = 77 \). Substitute this value into the equation:

\[ 77 = \frac{1}{2} n(n - 3) \]

Multiply both sides by 2 to eliminate the fraction:

\[ 154 = n(n - 3) \]

Expand the right side:

\[ 154 = n^2 - 3n \]

Rearrange the equation to form a standard quadratic equation:

\[ n^2 - 3n - 154 = 0 \]

To solve the quadratic equation \( n^2 - 3n - 154 = 0 \), we can use the quadratic formula:

\[ n = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]

where \( a = 1 \), \( b = -3 \), and \( c = -154 \).

Calculate the discriminant:

\[ b^2 - 4ac = (-3)^2 - 4 \times 1 \times (-154) = 9 + 616 = 625 \]

Calculate the roots:

\[ n = \frac{-(-3) \pm \sqrt{625}}{2 \times 1} = \frac{3 \pm 25}{2} \]

This gives two potential solutions:

\[ n = \frac{3 + 25}{2} = 14 \quad \text{and} \quad n = \frac{3 - 25}{2} = -11 \]

Since \( n \) must be a positive integer (a polygon cannot have a negative number of sides), we discard \( n = -11 \).

Thus, the valid solution is \( n = 14 \).

Final Answer