Question

If a convex polygon has $35$ diagonals, then the number of points of intersection of diagonals which lies inside the polygon is

• A. $45$
• B. $120$
• C. $210$
• D. $235$

Answer 1

The correct option is C $210$

If polygon has $n$ sides, then number of
diagonals is
${}^n{C}_2-n=35$
$\Rightarrow \frac{n(n-1)}{2}-n=35$
Solving we get $n=10.$
Thus, there are $10$ vertices (as number of sides and vertices in a polygon are always equal).

Four vertices can be selected in ${}^{10}{C}_4=210$ ways.

Using these four vertices two diagonals can be formed,
which has exactly one point of intersection lying inside the polygon.

$\therefore$number of points of intersections of diagonals which lies inside the polygon
$={}^{10}{C}_4\times 1=210.$