Answer:
C. 5
Step-by-step explanation:
We are given that the polygon has total 5 diagonals.
It is known that the formula for the number of diagonals in a polygon is,
Number of diagonals = [tex]\frac{n(n-3)}{2}[/tex], where n is the number of sides.
As, the total number of diagonals in the given polygon are 5.
So, we get,
5 = [tex]\frac{n(n-3)}{2}[/tex]
i.e. [tex]10=n^{2}-3n[/tex]
i.e. [tex]n^{2}-3n-10=0[/tex]
Since, the solution of a quadratic equation [tex]ax^2+bx+c=0[/tex] is given by, [tex]x=\frac{-b\pm \sqrt{b^{2}-4ac}}{2a}[/tex]
We have,
[tex]n^{2}-3n-10=0[/tex] implies a= 1, b= -3 and c= -10.
So, the solution is,
[tex]n=\frac{3\pm \sqrt{(-3)^{2}-4\times 1\times (-10)}}{2\times 1}[/tex]
i.e. [tex]n=\frac{3\pm \sqrt{9+40}}{2}[/tex]
i.e. [tex]n=\frac{3\pm \sqrt{49}}{2}[/tex]
i.e. [tex]n=\frac{3\pm 7}{2}[/tex]
i.e. [tex]n=\frac{3+7}{2}[/tex] and i.e. [tex]x=\frac{3-7}{2}[/tex]
i.e. [tex]n=\frac{10}{2}[/tex] and i.e. [tex]x=\frac{-4}{2}[/tex]
i.e. n= 5 and n= -2.
As, the number of sides cannot be negative.
So, n= 5.
Thus, the polygon will have 5 sides.
