Question

The angles of a triangle are as $1:2:7$, then the ratio of the greatest side to the least side is $\sqrt{5}+1:\sqrt{5}-1.$

Answer 1

Ans. $True$. Let the angle of triangle $\alpha ,2\alpha$ and $7\alpha$ so that $\alpha +2\alpha +7\alpha ={180}^o$, giving $\alpha ={18}^o$.
$\therefore$ Angle of the triangle are $A={18}^o$, $B={36}^o$ and $V={126}^o$, Then a is least and $c$, the greatest
side. Hence
$\frac{a}{\sin A}=\frac{c}{\sin C}\Rightarrow \frac{a}{\sin {18}^o}=\frac{c}{\sin {126}^o}$
$\Rightarrow \frac{c}{a}=\frac{\sin {126}^o}{\sin {18}^o}=\frac{\cos {36}^o}{\sin {18}^o}=\frac{(\sqrt{5}+1)/4}{(\sqrt{5}-1)/4}=\frac{\sqrt{5}+1}{\sqrt{5}-1}$