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,

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