Question

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

Answer 1

The number of intersecting diagonals can be found by ${}^n{C}_4$

or $n(n-1)(n-2)(n-3)=70\times 24=1680$

$n=8$

Now number of diagonals $={}^8{C}_2-8$

$\downarrow$

$8$ will be collinear

$\Rightarrow 28-8=20.$

Hence, the answer is $20$