Question

If in any triangle the angles be to one onother as $1:2:3$, prove that the corresponding sides are as $1:\sqrt{3}:2$,

Answer 1

Let $A,2A,3A$ be the angles

Then $A+2A+3A={180}^o$ or $A={30}^o$

The angle are ${30}^o$, ${60}^o$ and ${90}^o$.

If $a,b,c$ denote the sides opposite to these angles,

then

$\frac{a}{\sin {30}^o}=\frac{b}{\sin {60}^o}=\frac{c}{\sin {90}^o}$

$\therefore a:b:c=\sin {30}^o:\sin {60}^o:\sin {90}^o$

$=1/2:\sqrt{3}/2:1=1:\sqrt{3}:2$