If the angles of a triangle are in the ratio 1 : 2 : 7, then the ratio of the greatest side to the least side, is

Solution:

Let the angles of triangle ABC be $A=\theta , B = 2\theta$ and $C=7\theta .$ Then,

$\begin{array}{l}A+B+C={180}^{\circ }\Rightarrow 10\theta ={180}^{\circ }\Rightarrow \theta ={18}^{\circ }\\ \therefore A={18}^{\circ },B={36}^{\circ }\text{ and }C={126}^{\circ }\end{array}$

Clearly, c is the greatest side and a is the smallest side. 

Now,

$\begin{array}{l}\frac{a}{\sin A}=\frac{c}{\sin C}\\ \Rightarrow \frac{c}{a}=\frac{\sin C}{\sin A}=\frac{\sin {126}^{\circ }}{\sin {18}^{\circ }}=\frac{\cos {36}^{\circ }}{\sin {18}^{\circ }}=\frac{\sqrt{5}+1}{\sqrt{5}-1}\end{array}$