If the angles of a triangle A B C are in the ratio 1 :   2 :   3   , then the corresponding…

Given that angles of a triangle are in the ratio $1:2:3$

$\because$From angle sum property, we have, sum of all angles of a triangle $=180°$

And angle ratio is $1:2:3$

Hence, angles are $30°, 60°, 90°$

respectively.

So, according to sine rule

$\frac{a}{\sin A}=\frac{{\displaystyle b}}{{\displaystyle \sin B}}=\frac{{\displaystyle c}}{{\displaystyle \sin C}}$

$\Rightarrow \frac{a}{\sin 30°}=\frac{{\displaystyle b}}{{\displaystyle \sin 60°}}=\frac{{\displaystyle c}}{{\displaystyle \sin 90°}}$

$\Rightarrow \frac{a}{{\displaystyle \frac{1}{2}}}=\frac{{\displaystyle b}}{{\displaystyle \frac{{\displaystyle \sqrt{3}}}{{\displaystyle 2}}}}=\frac{{\displaystyle c}}{{\displaystyle 1}}$

$\Rightarrow \frac{a}{{\displaystyle 1}}=\frac{{\displaystyle b}}{{\displaystyle \sqrt{3}}}=\frac{{\displaystyle c}}{{\displaystyle 2}}$

Hence, ratio of $a:b:c$ will be $1:\sqrt{3}:2$