Question

If the angles of a triangle are in the ratio $2:3:7,$ then the sides opposite to these angles are in the ratio

• A. $\sqrt{2}:2:(\sqrt{3}+1)$
• B. $2:\sqrt{2}:(\sqrt{3}+1)$
• C. $1:\sqrt{2}:\frac{\sqrt{2}}{(\sqrt{3}-1)}$
• D. $\frac{1}{\sqrt{2}}:1:\left(\frac{\sqrt{3}+1}{2}\right)$

Answer 1

Answer: (A), (C), (D)

The correct options are
A $\sqrt{2}:2:(\sqrt{3}+1)$
C $1:\sqrt{2}:\frac{\sqrt{2}}{(\sqrt{3}-1)}$
D $\frac{1}{\sqrt{2}}:1:\left(\frac{\sqrt{3}+1}{2}\right)$

Let $A=2\alpha ,B=3\alpha ,C=7\alpha$
$\therefore A+B+C={180}^0$
$\Rightarrow 2\alpha +3\alpha +7\alpha ={180}^0$
$\Rightarrow 12\alpha ={180}^0$
$\Rightarrow \alpha ={15}^0$
$\therefore A=2\alpha =2\times {15}^0={30}^0$
$B=3\alpha =3\times {15}^0={45}^0$
$C=7\alpha =7\times {15}^0={105}^0$
$a:b:c=\sin {30}^0:\sin {45}^0:\sin {105}^0$
$=\frac{1}{2}:\frac{1}{\sqrt{2}}:\frac{\sqrt{3}+1}{2\sqrt{2}}$
$a:b:c=\sqrt{2}:2:\sqrt{3}+1$
$=\frac{1}{\sqrt{2}}:1:\frac{\sqrt{3}+1}{2}$
or $1:\sqrt{2}:\frac{\sqrt{2}(\sqrt{3}+1)(\sqrt{3}-1)}{2(\sqrt{3}-1)}$
or $1:\sqrt{2}:\frac{2\sqrt{2}}{2(\sqrt{3}-1)}$
$\therefore a:b:c=1:\sqrt{2}:\frac{\sqrt{2}}{\sqrt{3}-1}$