Question

A regular polygon of $n$ sides has $170$ diagonals. Then $n = ?$

Answer 1

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.