Question

In a polygon no three diagonals are concurrent except at vertices. If the total number of points of intersection of diagonals interior to the polygon be 70, then the number of diagonals of the polygon is

• A. 20
• B. 28
• C. 8
• D. 56

Answer 1

The correct option is A 20

Let number of sides = n
A selection of four vertices of the polygon gives an interior intersection
${=}^n{C}_4=70\Rightarrow n(n-1)(n-2)(n-3)$
$=24\times 70=8\times 7\times 6\times 5\Rightarrow n=8$
Number of diagonals =${}^n{C}_2-n{=}^8{C}_2-8=28-8=20$