Permutations & Combinations
Grade None
Question:
<p>If in a regular polygon the number of diagonals is 54, then the number of sides of this polygon is</p>
<p style="display:inline">6</p>
<p style="display:inline">12</p>
<p style="display:inline">10</p>
<p style="display:inline">9</p>
Step-by-Step Solution
Key Concept: The number of diagonals in an n-sided polygon is determined by the formula n(n-3)/2, which subtracts the n boundary sides from the total possible line segments connecting n vertices.
<p>Number of diagonal = 54<br />
<span class="math-tex">$\Rightarrow \frac{n(n-3)}{2}$</span> = 54<br />
<span class="math-tex">$\Rightarrow$</span> n<sup>2</sup> - 3n - 108 = 0<br />
<span class="math-tex">$\Rightarrow$</span> n<sup>2</sup> - 12n + 9n - 108 = 0<br />
<span class="math-tex">$\Rightarrow$</span> n = (n - 12)+ 9 (n - 12) = 0<br />
<span class="math-tex">$\Rightarrow$</span> n = 12, -9<br />
<span class="math-tex">$\Rightarrow$</span> n = 12 (<span class="math-tex">$\because$</span> n <span class="math-tex">$\neq$</span> -9)</p>
Correct Answer: B