The angles of a triangle are in the ratio 3:5:10. Then the ratio of the smallest side to the greatest side is

Angles are in the ratio

 $$\begin{gathered} A:B:C=3:5:10\Rightarrow A=3k,B=5k, C=10k \\ and A+B+C=180°\Rightarrow 3k+5k+10k=180°\Rightarrow k=10° \\ \therefore \text{ Angles are }{30}^{\circ },{50}^{\circ },{100}^{\circ }\text{. } \end{gathered}$$

 Smallest side: Greatest side
$$\begin{gathered} a:c=2R\sin {30}^{\circ }:2R\sin {100}^{\circ }=2R\sin {30}^{\circ }:2R\sin {100}^{\circ } \\ \Rightarrow a:c=\frac{1}{2}:\cos {10}^{\circ }=1:2\cos {10}^{\circ } \end{gathered}$$.