Question

If the angles of a triangle are in the ratio $4:1:1,$ then the ratio of the longest side to the perimeter is

• A. $\sqrt{3}:(2+\sqrt{3})$
• B. $1:6$
• C. $1:(2+\sqrt{3})$
• D. $2:3$

Answer 1

The correct option is A $\sqrt{3}:(2+\sqrt{3})$

Let $\angle A=4x,\angle B=x,\angle C=x$

$\Rightarrow 6x={180}^0\Rightarrow x={30}^0$

$\Rightarrow \angle A={120}^0,\angle B={30}^0,\angle C={30}^0$

$\therefore$ By sine formula $a=k\frac{\sqrt{3}}{2},b=\frac{k}{2},c=\frac{k}{2}$

$\therefore$ Longest side $:$ perimeter $=\frac{\frac{\sqrt{3}}{2}}{(\frac{\sqrt{3}}{2}+1)}=\frac{\sqrt{3}}{2+\sqrt{3}}$