Question

If the number of diagonals of a regular polygon is 35, then the number of sides of the polygon is

• A. \( 12 \)
• B. \( 9 \)
• C. \( 10 \)
• D. \( 11 \) Correct Answer

Hint:

The number of diagonals in a polygon with \(n\) sides is \( D = \frac{n(n-3)}{2} \). This formula arises from choosing any two vertices to form a line segment (\( {}^n C_2 \) ways) and then subtracting the \(n\) sides of the polygon. Remember that \(n\) must be a positive integer greater than or equal to 3.

Answer 1

The Correct Option is C

Step 1: Recall the formula for the number of diagonals in a polygon.
The number of diagonals \(D\) in a polygon with \(n\) sides is given by the formula: \[ D = \frac{n(n-3)}{2} \]
Step 2: Substitute the given number of diagonals into the formula.
We are given that \( D = 35 \).
\[ \frac{n(n-3)}{2} = 35 \]
Step 3: Solve the equation for \(n\).
\[ n(n-3) = 35 \times 2 \] \[ n(n-3) = 70 \] \[ n^2 - 3n = 70 \] \[ n^2 - 3n - 70 = 0 \] This is a quadratic equation in \(n\).
We can solve it by factoring or using the quadratic formula.
To factor, we look for two numbers that multiply to -70 and add to -3.
These numbers are -10 and 7.
\[ (n-10)(n+7) = 0 \] This gives two possible values for \(n\): \( n-10=0 \Rightarrow n=10 \) or \( n+7=0 \Rightarrow n=-7 \).

Step 4: Choose the valid value for \(n\).
Since \(n\) represents the number of sides of a polygon, it must be a positive integer, and \(n \ge 3\).
Therefore, \( n=10 \) is the valid solution.
The number of sides of the polygon is 10.
This matches option (3).