Question

How many diagonals are there in a $50$-sided polygon ?

Answer 1

Hint: A polygon with $n$ sides should also have $n$ vertices. We can get either a side of a polygon or diagonal of a polygon by joining any two vertices. With the help of permutation in calculation, we take $2$ points at a time and we get the number of lines joining all the points, then subtracting the number of edges and then, we get the total number of diagonals.

Complete step by step solution:
Given is a $50$-sided polygon. We have to find out the number of diagonals in a $50$-sided polygon.The number of line segments can only be obtained by joining the vertices of a $50$-sided polygon taking two points at a time. Now, we apply the formula and use permutation to get our solution. The number of ways of selecting $2$ points at a time from a $n$ number of points is given as ${}^n{C_2}$.
We have, ${}^n{C_r} = \dfrac{{n!}}{{r!\left( {n - r} \right)!}}$
${}^n{C_2} = \dfrac{{n!}}{{2!\left( {n - 2} \right)!}}$
On further simplification,
${}^n{C_2} = \dfrac{{n\left( {n - 1} \right)\left( {n - 2} \right)!}}{{2!\left( {n - 2} \right)!}} \\
\Rightarrow \dfrac{{n\left( {n - 1} \right)}}{2} \\ $
Hence, out of the total selections here$n$are the sides of polygon so subtracting that from total selections we get,
$\dfrac{{n\left( {n - 1} \right)}}{2} - n \\
\Rightarrow \dfrac{{n\left( {n - 1} \right) - 2n}}{2} \\
\Rightarrow \dfrac{{n\left( {n - 1 - 2} \right)}}{2} \\
\Rightarrow \dfrac{{n\left( {n - 3} \right)}}{2} \\ $
Therefore, there are a total $\dfrac{{n\left( {n - 3} \right)}}{2}$ number of diagonals in an $n$ sided polygon. According to the question, we are given $n = 50$. So, the number of diagonals in a $50$-sided polygon will be:
$\dfrac{{50\left( {50 - 3} \right)}}{2} \\
\Rightarrow \dfrac{{50 \times 47}}{2} \\
\Rightarrow 25 \times 47 \\
\Rightarrow 1175 \\ $
Hence, there are $1175$ diagonals in a $50$-sided polygon.

Note: In geometrical terms, a plane figure which is described by a finite number of line segments connected to form a closed polygon chain or polygonal circuit is called a polygon. A line segment which joins two vertices of a polygon is called a diagonal. Simply, any sloping line is known as diagonal.