Question

The sides of a triangle are three consecutive natural numbers and its largest angle is twice the smallest one. Then the sides of the triangles are

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

Answer 1

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

Let $a=n-1,b=n,c=n+1;n\in N$ and $n\ge 2$

$\therefore$ larger angle $=C$ and smallest angle $=A$

Since the sides of triangle are in $A.P.$, then $C=2A$

$\sin C=\sin 2A\Rightarrow \sin C=2\sin A\cos A$

$\Rightarrow \frac{c}{2a}=\cos A\Rightarrow \frac{b^2+c^2-a^2}{2bc}=\frac{c}{2a}$

$\Rightarrow \frac{\left(n^2\right){(n+1)}^2-{(n-1)}^2}{\left(n\right)(n+1)}=\frac{n+1}{2(n-1)}$

$\Rightarrow n=5\Rightarrow a=4,b=5,c=6$