Answer
Verified
388.2k+ views
Hint: A polygon of n sides should also be having n vertices. By joining any two vertices of a polygon, we obtain either a side of that polygon or a diagonal of that polygon. So calculating with the help of permutation, by taking \[{\text{2}}\] points at a time, we get the number of lines joining all the points, then subtracting the number of edges we get the total number of diagonals.
Complete step by step answer:
We have to find the numbers of diagonals in the n-sided polygon.
The number of line segments obtained by joining the vertices of a n sided polygon taken two points at a time.
Now, applying the formula and using permutation as below stated.
The number of ways of selecting 2 points at a time from n number of points is given as \[^{\text{n}}{{\text{C}}_{\text{2}}}\]
As we have \[^{\text{n}}{{\text{C}}_r} = \dfrac{{n!}}{{r!\left( {n - r} \right)!}}\]
So we have
\[^{\text{n}}{{\text{C}}_{\text{2}}}{\text{ = }}\dfrac{{{\text{n!}}}}{{{\text{2!}}\left( {{\text{n - 2}}} \right){\text{!}}}}\]
On simplifying we get,
\[{ \Rightarrow ^{\text{n}}}{{\text{C}}_{\text{2}}}{\text{ = }}\dfrac{{{\text{n(n - 1)(n - 2)!}}}}{{{\text{2!}}\left( {{\text{n - 2}}} \right){\text{!}}}}\]
\[ \Rightarrow \dfrac{{{\text{n(n - 1)}}}}{{\text{2}}}\]
Hence, out of the total selections here n are the sides of the polygon so subtracting that from the total selections, we get,
\[ \Rightarrow \dfrac{{{\text{n(n - 1)}}}}{{\text{2}}} - n\]
On simplifying we get,
\[ \Rightarrow \dfrac{{{\text{n(n - 1) - 2n}}}}{{\text{2}}}\]
On taking n common from both the terms we get,
\[
\Rightarrow \dfrac{{{\text{n(n - 1 - 2)}}}}{{\text{2}}} \\
\Rightarrow \dfrac{{{\text{n(n - 3)}}}}{{\text{2}}} \\
\]
Hence , there are total \[\dfrac{{{\text{n(n - 3)}}}}{{\text{2}}}\] number of diagonals in an n sided polygon.
Note:: Don’t forget to subtract the number of sides while finding the number of diagonals. In geometry, a polygon is a plane figure that is described by a finite number of straight-line segments connected to form a closed polygonal chain or polygonal circuit. The solid plane region, the bounding circuit, or the two together, may be called a polygon. In geometry, a diagonal is a line segment joining two vertices of a polygon or polyhedron, when those vertices are not on the same edge. Informally, any sloping line is called diagonal.
Complete step by step answer:
We have to find the numbers of diagonals in the n-sided polygon.
The number of line segments obtained by joining the vertices of a n sided polygon taken two points at a time.
Now, applying the formula and using permutation as below stated.
The number of ways of selecting 2 points at a time from n number of points is given as \[^{\text{n}}{{\text{C}}_{\text{2}}}\]
As we have \[^{\text{n}}{{\text{C}}_r} = \dfrac{{n!}}{{r!\left( {n - r} \right)!}}\]
So we have
\[^{\text{n}}{{\text{C}}_{\text{2}}}{\text{ = }}\dfrac{{{\text{n!}}}}{{{\text{2!}}\left( {{\text{n - 2}}} \right){\text{!}}}}\]
On simplifying we get,
\[{ \Rightarrow ^{\text{n}}}{{\text{C}}_{\text{2}}}{\text{ = }}\dfrac{{{\text{n(n - 1)(n - 2)!}}}}{{{\text{2!}}\left( {{\text{n - 2}}} \right){\text{!}}}}\]
\[ \Rightarrow \dfrac{{{\text{n(n - 1)}}}}{{\text{2}}}\]
Hence, out of the total selections here n are the sides of the polygon so subtracting that from the total selections, we get,
\[ \Rightarrow \dfrac{{{\text{n(n - 1)}}}}{{\text{2}}} - n\]
On simplifying we get,
\[ \Rightarrow \dfrac{{{\text{n(n - 1) - 2n}}}}{{\text{2}}}\]
On taking n common from both the terms we get,
\[
\Rightarrow \dfrac{{{\text{n(n - 1 - 2)}}}}{{\text{2}}} \\
\Rightarrow \dfrac{{{\text{n(n - 3)}}}}{{\text{2}}} \\
\]
Hence , there are total \[\dfrac{{{\text{n(n - 3)}}}}{{\text{2}}}\] number of diagonals in an n sided polygon.
Note:: Don’t forget to subtract the number of sides while finding the number of diagonals. In geometry, a polygon is a plane figure that is described by a finite number of straight-line segments connected to form a closed polygonal chain or polygonal circuit. The solid plane region, the bounding circuit, or the two together, may be called a polygon. In geometry, a diagonal is a line segment joining two vertices of a polygon or polyhedron, when those vertices are not on the same edge. Informally, any sloping line is called diagonal.
Recently Updated Pages
The branch of science which deals with nature and natural class 10 physics CBSE
The Equation xxx + 2 is Satisfied when x is Equal to Class 10 Maths
Define absolute refractive index of a medium
Find out what do the algal bloom and redtides sign class 10 biology CBSE
Prove that the function fleft x right xn is continuous class 12 maths CBSE
Find the values of other five trigonometric functions class 10 maths CBSE
Trending doubts
Difference Between Plant Cell and Animal Cell
Difference between Prokaryotic cell and Eukaryotic class 11 biology CBSE
How do you solve x2 11x + 28 0 using the quadratic class 10 maths CBSE
Select the correct plural noun from the given singular class 10 english CBSE
What organs are located on the left side of your body class 11 biology CBSE
The sum of three consecutive multiples of 11 is 363 class 7 maths CBSE
What is the z value for a 90 95 and 99 percent confidence class 11 maths CBSE
Fill the blanks with the suitable prepositions 1 The class 9 english CBSE
How many squares are there in a chess board A 1296 class 11 maths CBSE