Given that
The angles of the triangle are
30, 60 and 90
Consider the sine law formula
[tex]\frac{a}{\sin A}=\frac{b}{\sin B}=\frac{c}{\sin C}[/tex]Let A=30, B=60, and C=90, and substitute these values in the sine law, we get
[tex]\frac{a}{\sin 30^o}=\frac{b}{\sin 60^o}=\frac{c}{\sin 90^o}[/tex][tex]\frac{a}{\frac{1}{2}}^{}=\frac{b}{\frac{\sqrt[]{3}}{2}}=\frac{c}{1}[/tex]The ratio of the legs a and b is
[tex]\frac{a}{b}=\frac{\frac{1}{2}}{\frac{\sqrt[]{3}}{2}}=\frac{1}{2}\times\frac{2}{\sqrt[]{3}}=\frac{1}{\sqrt[]{3}}[/tex][tex]1\colon\sqrt[]{3}[/tex]The ratio of the legs b and c is
[tex]\frac{b}{c}=\frac{\frac{\sqrt[]{3}}{2}}{1}=\frac{\sqrt[]{3}}{2}[/tex][tex]\sqrt[]{3}\colon2[/tex]The ratio of the legs a and c is
[tex]\frac{a}{c}=\frac{\frac{1}{2}}{1}=\frac{1}{2}[/tex][tex]1\colon2[/tex]Hence the answer is
[tex]1\colon\sqrt[]{3}[/tex]Option C is correct.