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

The correct option is B $4,5,6$

Let the sides of triangle are $x-1,x,x+1$
From sine rule
$\frac{\sin A}{A}=\frac{\sin B}{B}=\frac{\sin C}{C}$
$\Rightarrow \frac{\sin \theta }{x-1}=\frac{\sin 2\theta }{x+1}$
$\Rightarrow \frac{\sin \theta }{x-1}=\frac{2\sin \theta \cos \theta }{x+1}$
$\Rightarrow x+1-2\cos \theta (x-1)=\mathrm{0...}\left(1\right)$
By cosine rule
$\cos \theta =\frac{(x+1{)}^2+x^2-(x-1{)}^2}{2x(x+1)}$
$\Rightarrow \cos \theta =\frac{x+4}{2(x+1)}...\left(2\right)$
From $\left(1\right)$ and $\left(2\right)$
$x+1-2\left(\frac{x+4}{2(x+1)}\right)(x-1)=0$
$\Rightarrow x=5$
Hence, sides of triangle are $4,5,6$