Question

If in a regular polygon, the number of diagonals are 54, then the number of sides of the polygon are}

• A. 10
• B. 12
• C. 9
• D. 6

Hint:

Remember the formula for the number of diagonals in a polygon with $n$ sides: $\frac{n(n-3)}{2}$. This formula is derived by considering all possible line segments between $n$ vertices and subtracting the $n$ sides.

Answer 1

The Correct Option is A

Step 1: State the formula for the number of diagonals. The number of diagonals in a polygon with $n$ sides is given by the formula: \[N_d = \frac{n(n-3)}{2}\] Alternatively, it can be thought of as selecting 2 vertices out of $n$ vertices ($^nC_2$) and subtracting the $n$ sides of the polygon, so $N_d = {^nC_2} - n = \frac{n(n-1)}{2} - n$.
Step 2: Set up the equation using the given number of diagonals. We are given that the number of diagonals is 54. Using the formula $N_d = \frac{n(n-3)}{2}$, we set up the equation: \[\frac{n(n-3)}{2} = 54\]
Step 3: Solve the quadratic equation for $n$. Multiply both sides by 2: \[n(n-3) = 108\] Expand the left side: \[n^2 - 3n = 108\] Rearrange into a standard quadratic equation form: \[n^2 - 3n - 108 = 0\] Factor the quadratic equation. We need two numbers that multiply to -108 and add to -3. These numbers are -12 and 9. \[(n - 12)(n + 9) = 0\] This gives two possible values for $n$: \[n - 12 = 0 \Rightarrow n = 12\] \[n + 9 = 0 \Rightarrow n = -9\]
Step 4: Choose the valid value for $n$. The number of sides of a polygon cannot be negative. Therefore, the valid number of sides is $n = 12$.