<p>If in a regular polygon the number of diagonals is 54, then the number of sides of this polygon is</p>
Step-by-Step Solution
Key Concept: A diagonal connects two non-adjacent vertices. For an n-sided polygon, total vertex pairs = C(n,2), but we exclude n sides, giving diagonals = n(n-3)/2. Set this equal to 54 and solve for n.
<p><strong>Step 1:</strong> In a regular n-sided polygon, the number of diagonals is given by the formula:</p><p>Number of diagonals = C(n,2) − n = n(n−1)/2 − n</p><p><strong>Step 2:</strong> Simplify the formula:</p><p>= n(n−1)/2 − 2n/2 = [n(n−1) − 2n]/2 = n(n−3)/2</p><p><strong>Step 3:</strong> Set equal to 54:</p><p>n(n−3)/2 = 54</p><p>n(n−3) = 108</p><p>n² − 3n − 108 = 0</p><p><strong>Step 4:</strong> Solve using the quadratic formula or factoring:</p><p>(n − 12)(n + 9) = 0</p><p>n = 12 or n = −9</p><p><strong>Step 5:</strong> Since n must be positive, n = 12.</p><p><strong>Verification:</strong> Diagonals = 12(12−3)/2 = 12(9)/2 = 54 ✓</p><p>∴ Answer: <strong>12 sides</strong></p>
Correct Answer: C