Question 979366

If the angles are in a ratio of {{{1:2:3}}}, then let {{{x }}} be smallest angle.


The other angles are {{{2x }}}and {{{3x}}} respectively, so 
{{{x + 2x + 3x=180}}}
{{{6x=180}}}
{{{x=30}}}
{{{2x=60}}}
{{{3x=90}}}
{{{x=30}}}


so, this is a {{{30-60-90}}} triangle, so the sides are in a ratio of {{{1:2:sqrt(3)}}}.


Let 
{{{y}}} be first side
{{{2y}}} second side (hypotenuse of right triangle)
{{{y*sqrt(3)}}}  third side


Perimeter 
{{{ P=y+ 2y + y*sqrt(3)=30+10*sqrt(3)}}}
{{{3y+ y*sqrt(3) = 30+ 10*sqrt(3)}}}


Therefore, {{{y=10}}}.

the smallest side is {{{highlight(10)}}}, 
the second side is {{{20}}}, and 
the third side is {{{10*sqrt(3)}}}