Answer
Verified
402.9k+ views
Hint: We know that, the total number of lines joining any two points of the polygon of \[n\] sides are given by \[^n{C_2} = \dfrac{{n(n - 1)}}{2}\]
So, the number of diagonals of a polygon of \[n\] sides are \[\dfrac{{n(n - 1)}}{2} - n\]
Using this formula we are going to relate that is to equate the number of diagonals given and the formula, by equating we can solve and find the number of sides.
Complete step-by-step answer:
It is given that a regular polygon of \[n\] sides has \[170\] diagonals. We have to find the number of sides of the polygon.
We know that, the total number of lines joining any two points of the polygon of \[n\] sides are given by \[^n{C_2} = \dfrac{{n(n - 1)}}{2}\]
So, the number of diagonals of a polygon of \[n\] sides are \[\dfrac{{n(n - 1)}}{2} - n\]
According to the problem, there are 170 diagonals,
So we equate the 170 to the formula,
\[\dfrac{{n(n - 1)}}{2} - n = 170\]
Let us simplify the left hand side of the equation we get,
\[\dfrac{{{n^2} - n - 2n}}{2} = 170\]
Let us multiply both sides of the equation by 2 we get,
\[{n^2} - 3n = 340\]
Let us now subtract by 340 both sides we get,
\[{n^2} - 3n - 340 = 0\]
Now, we will apply the middle term factor rule.
\[{n^2} - 20n + 17n - 340 = 0\]
So, we have,
\[(n - 20)(n + 17) = 0\]
On equating both the terms of the product to zero we get two values.
In that we will take \[n = 20\] since we will not take the negative values for a number of sides.
Hence, the number of sides of the polygon is \[20.\]
Note: From the quadratic equation we get, two values of number of sides. We will take the positive value since the number of sides cannot be negative.
So, the number of diagonals of a polygon of \[n\] sides are \[\dfrac{{n(n - 1)}}{2} - n\]
Using this formula we are going to relate that is to equate the number of diagonals given and the formula, by equating we can solve and find the number of sides.
Complete step-by-step answer:
It is given that a regular polygon of \[n\] sides has \[170\] diagonals. We have to find the number of sides of the polygon.
We know that, the total number of lines joining any two points of the polygon of \[n\] sides are given by \[^n{C_2} = \dfrac{{n(n - 1)}}{2}\]
So, the number of diagonals of a polygon of \[n\] sides are \[\dfrac{{n(n - 1)}}{2} - n\]
According to the problem, there are 170 diagonals,
So we equate the 170 to the formula,
\[\dfrac{{n(n - 1)}}{2} - n = 170\]
Let us simplify the left hand side of the equation we get,
\[\dfrac{{{n^2} - n - 2n}}{2} = 170\]
Let us multiply both sides of the equation by 2 we get,
\[{n^2} - 3n = 340\]
Let us now subtract by 340 both sides we get,
\[{n^2} - 3n - 340 = 0\]
Now, we will apply the middle term factor rule.
\[{n^2} - 20n + 17n - 340 = 0\]
So, we have,
\[(n - 20)(n + 17) = 0\]
On equating both the terms of the product to zero we get two values.
In that we will take \[n = 20\] since we will not take the negative values for a number of sides.
Hence, the number of sides of the polygon is \[20.\]
Note: From the quadratic equation we get, two values of number of sides. We will take the positive value since the number of sides cannot be negative.
Recently Updated Pages
Assertion The resistivity of a semiconductor increases class 13 physics CBSE
The Equation xxx + 2 is Satisfied when x is Equal to Class 10 Maths
How do you arrange NH4 + BF3 H2O C2H2 in increasing class 11 chemistry CBSE
Is H mCT and q mCT the same thing If so which is more class 11 chemistry CBSE
What are the possible quantum number for the last outermost class 11 chemistry CBSE
Is C2 paramagnetic or diamagnetic class 11 chemistry CBSE
Trending doubts
Difference Between Plant Cell and Animal Cell
Difference between Prokaryotic cell and Eukaryotic class 11 biology CBSE
What is BLO What is the full form of BLO class 8 social science CBSE
Fill the blanks with the suitable prepositions 1 The class 9 english CBSE
Fill the blanks with proper collective nouns 1 A of class 10 english CBSE
Change the following sentences into negative and interrogative class 10 english CBSE
Give 10 examples for herbs , shrubs , climbers , creepers
What organs are located on the left side of your body class 11 biology CBSE
Name 10 Living and Non living things class 9 biology CBSE