<p>In a polygon, no three diagonals are concurrent. If the total number of points of intersection of diagonals interior to the polygon is 70, then the number of diagonals of the polygon is</p>
Step-by-Step Solution
Key Concept: Each intersection point of diagonals inside a polygon is uniquely determined by choosing 4 vertices (where the two diagonals connect opposite pairs). Thus, interior intersection points = C(n,4), and we solve C(n,4) = 70 to find n, then use diagonals = n(n-3)/2.
<p><strong>Step 1:</strong> Establish the relationship between intersection points and vertices.</p><p>Each interior intersection point of diagonals is formed by exactly 4 vertices of the polygon. If we choose any 4 vertices, they determine 2 diagonals that intersect inside the polygon (connecting opposite pairs in the quadrilateral formed).</p><p><strong>Step 2:</strong> Set up the equation using combinations.</p><p>Number of interior intersection points = C(n,4) = 70, where n is the number of sides.</p><p>C(n,4) = n(n-1)(n-2)(n-3)/24 = 70</p><p>Therefore: n(n-1)(n-2)(n-3) = 1680</p><p><strong>Step 3:</strong> Solve for n.</p><p>Testing values: For n = 8: 8 × 7 × 6 × 5 = 1680 ✓</p><p><strong>Step 4:</strong> Find the number of diagonals.</p><p>Number of diagonals = n(n-3)/2 = 8(8-3)/2 = 8(5)/2 = 20</p><p>∴ Answer: D (20 diagonals)</p>
Correct Answer: D