Question

If the angles of the 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 $\sqrt{3}:(2+\sqrt{3})$

Given, ratio of angles are $4:1:1$
$\Rightarrow 4y+y+y={180}^{\circ }\Rightarrow y={30}^{\circ }$
$\therefore \angle A={120}^{\circ },\angle B={30}^{\circ },\angle C={30}^{\circ }$
Thus, the ratio of longest side to perimeter $=\frac{a}{a+b+c}$
Let $b=c=x$
$a^2=b^2+c^2-2bc\cos A\Rightarrow a^2=2x^2-2x^2\cos A=2x^2(1-\cos A)\Rightarrow a=2x\sin \frac{A}{2}\Rightarrow a=2x\sin {60}^{\circ }=\sqrt{3}x$
Thus, required ratio is
$\frac{a}{a+b+c}=\frac{\sqrt{3}}{\sqrt{3}+2}$