Question

The sides of a triangle are three consecutive natural numbers and its largest angle is twice the smallest one, then the sides are

• A. $6$, $7$, $8$
• B. $4$, $5$, $6$
• C. $1$, $2$, $3$
• D. $3$, $4$, $5$

Answer 1

Answer: (B)

Using sine law,

$ \frac{\sin \,\alpha}{n-1}=\frac{\sin \,2 \alpha}{n+1} $
$ \Rightarrow \, 2 \cos \alpha=\frac{n+1}{(n-1)}$
$ \Rightarrow \, \cos \alpha=\frac{n+1}{2(n-1)} $
$\Rightarrow \, \frac{n^{2}+(n+1)^{2}-(n-1)^{2}}{2 n(n+1)} $
$ \Rightarrow \,\frac{(n+1)}{2(n-1)} $ (using cosine law)
$ \Rightarrow \, \frac{n^{2}+4 n}{2 n+(n+1)}=\frac{(n+1)}{2(n-1)} $
$ \Rightarrow \, \frac{n+4}{2(n+1)}=\frac{n+1}{2(n-1)} $
$ \therefore \,(n+1)^{2}=(n+4)(n-1)$
$\Rightarrow \, n=5 $
Hence, lengths of the side of the triangle are 4,5 and $6$