Question

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

• A. $12$
• B. $17$
• C. $20$
• D. $25$

Answer 1

Answer: (C)

$20$

Given: The polygon has 170 diagonals.

To Find: The number of sides.

Let us assume that the number of sides of the polygon are be $n$

The total number of lines joining any two points of the polygon is given by ${}^n{C}_2=\frac{n\times (n-1)}{2}$.

Now the number of sides in a polygon is $n$.

Hence the number of diagonals in a polygon of $n$ sides is given by $\frac{n\times (n-1)}{2}-n$.

Hence, $170=\frac{n\times (n-1)}{2}-n$

$170=\frac{n^2-n-2n}{2}$

$n^2-3n-340=0$

The solution to this is given by,

$n=\frac{3\pm \sqrt{9+4\times 340}}{2}$

$n=\frac{3\pm 37}{2}$

Hence, $n=20.$