Question

Let P be a polygon with \(n\) vertices such that the line segment joining any two points of P remains entirely in P. If the number of diagonals of P is \(n+25\), then \(n\) is equal to ________.

Hint:

The polygon is convex, so the number of diagonals is \(\frac{n(n-3)}{2}\).

Answer 1

Correct Answer: 10

A polygon in which the segment joining any two of its points stays inside it is convex. A convex polygon with \(n\) vertices has \({}^{n}C_2-n=\frac{n(n-3)}{2}\) diagonals.
Set \(\frac{n(n-3)}{2}=n+25\Rightarrow n^2-3n=2n+50\Rightarrow n^2-5n-50=0\Rightarrow (n-10)(n+5)=0\).
Since \(n>0\), \(n=10\).
Answer: 10