Question

The ratio between the angles of $a$ $△$ is $4:1:1$. Find the ratio between its greatest side and its perimeter.

Answer 1

Given Ratio between angles of a $△$ is $4:1:1$

Let the $△$ be $△ABC$ and angles be $A=4x,B=x,C=x$

In $△ABC,A+B+C=\pi$

$⟹4x+x+x=\pi ⟹x=\frac{\pi }{6}$

$A=\frac{2\pi }{3},B=\frac{\pi }{6},C=\frac{\pi }{6}$

$⟹a=2R\sin A=2R\sin \frac{2\pi }{3}=\sqrt{3}R,b=c=2R\sin B=2R\sin \frac{\pi }{6}=R$

Greatest Side is $a=\sqrt{3}R$

Perimeter $=a+b+c=\sqrt{3}R+R+R=(\sqrt{3}+2)R$

Ratio between greatest side and perimeter $=\sqrt{3}R:(\sqrt{3}+2)R=\sqrt{3}:(\sqrt{3}+2)$