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A regular polygon of $n$ sides has 170 diagonals, then $n$ is equal to
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Verified Answer
The correct answer is:
20
A polygon of $n$ sides has number of diagonals $=170$
$$
\begin{array}{ll}
\Rightarrow & \frac{n(n-3)}{2}=170 \\
\Rightarrow & n^2-3 n=340 \\
\Rightarrow & n^2-3 n-340=0 \\
\Rightarrow & n^2-20 n+17 n-340=0 \\
\Rightarrow & n(n-20)+17(n-20)=0 \\
\Rightarrow & (n-20)(n+17)=0 \\
\Rightarrow & n-20=0 \text { or } n+17=0 \\
& n=20 \text { or } n=-17 \quad \text { [it is not possible] }
\end{array}
$$
So, $n=20$.
$$
\begin{array}{ll}
\Rightarrow & \frac{n(n-3)}{2}=170 \\
\Rightarrow & n^2-3 n=340 \\
\Rightarrow & n^2-3 n-340=0 \\
\Rightarrow & n^2-20 n+17 n-340=0 \\
\Rightarrow & n(n-20)+17(n-20)=0 \\
\Rightarrow & (n-20)(n+17)=0 \\
\Rightarrow & n-20=0 \text { or } n+17=0 \\
& n=20 \text { or } n=-17 \quad \text { [it is not possible] }
\end{array}
$$
So, $n=20$.
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