Question

Consider the following statements:
Statement I : The number of diagonals of $n$-sided polygon is ${ }^{n} C_{2}-n$
Statement II : A polygon has $44$ diagonals. The number of its sides are $10$.
Choose the correct option from the choices given below.

• A. Only I is true
• B. Only II is true
• C. Both I and II are true
• D. Both I and II are false

Answer 1

Answer: (A)

I. In $n$-sided polygon, the number of vertices $= n$
$\therefore $ Number of lines that can be formed using $n$ points $={ }^{ n } C _{2}$
Out of these, ${ }^{n} C _{2}$ lines, $n$ lines from the polygon.
$\therefore $ Number of diagonals $={ }^{ n } C _{2}- n$
II. Let the number of sides of a polygon $= n$
Number of diagonal = Number of line segment joining any two vertices of polygon $-$ Number of sides
$={ }^{ h } C _{2}- n$
$=\frac{n(n-1)}{2}-n=\frac{n(n-3)}{2}$
Now, $\frac{ n ( n -3)}{2}=44 $
$\Rightarrow n ^{2}-3 n -88=0$
$\Rightarrow (n-11)(n+8)=0 $
$\Rightarrow n=11$
or $n =-8$ rejected.