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

Question

Aicharya · Accepted Answer

Let the angles of the triangle be $4 \dot{x}, x$ and $x$. $\begin{aligned} 	herefore \quad & 4 x+x+x=180^{\circ} \Rightarrow 6 x=180^{\circ} \Rightarrow x=30^{\circ} \ & \frac{\sin 120^{\circ}}{\mathrm{a}}=\frac{\sin 30^{\circ}}{\mathrm{b}}=\frac{\sin 30^{\circ}}{\mathrm{c}} \ 	herefore \quad & \mathrm{a}:(\mathrm{a}+\mathrm{b}+\mathrm{c}) \ & =\left(\sin 120^{\circ}ight):\left(\sin 120^{\circ}+\sin 30^{\circ}+\sin 30^{\circ}ight) \ \quad & \frac{\sqrt{3}}{2}: \frac{\sqrt{3}+2}{2}=\sqrt{3}: \sqrt{3}+2 \end{aligned}$

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

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