Search any question & find its solution
Question:
Answered & Verified by Expert
If in a regular polygon, the number of diagonals are 54 , then the number of sides of the polygon are
Options:
Solution:
2888 Upvotes
Verified Answer
The correct answer is:
12
Number of diagonals in a polygon of $\mathrm{n}$ sides is ${ }^{\mathrm{n}} \mathrm{C}_2-\mathrm{n}$.
$\begin{aligned}
\therefore \quad & { }^{\mathrm{n}} \mathrm{C}_2-\mathrm{n}=54 \\
& \Rightarrow \frac{\mathrm{n}(\mathrm{n}-1)}{2}-\mathrm{n}=54 \\
& \Rightarrow \mathrm{n}^2-3 \mathrm{n}-108=0 \\
& \Rightarrow(\mathrm{n}-12)(\mathrm{n}+9)=0
\end{aligned}$
But, number of sides cannot be negative.
$\therefore \quad \mathrm{n}=12$
$\begin{aligned}
\therefore \quad & { }^{\mathrm{n}} \mathrm{C}_2-\mathrm{n}=54 \\
& \Rightarrow \frac{\mathrm{n}(\mathrm{n}-1)}{2}-\mathrm{n}=54 \\
& \Rightarrow \mathrm{n}^2-3 \mathrm{n}-108=0 \\
& \Rightarrow(\mathrm{n}-12)(\mathrm{n}+9)=0
\end{aligned}$
But, number of sides cannot be negative.
$\therefore \quad \mathrm{n}=12$
Looking for more such questions to practice?
Download the MARKS App - The ultimate prep app for IIT JEE & NEET with chapter-wise PYQs, revision notes, formula sheets, custom tests & much more.