Question 978667
You could do the final computation last, and just go with {{{p=30+10sqrt(3)}}}.


x, common factor in each of the angles.
{{{1x+2x+3x=180}}}
{{{6x=180}}}
{{{x=180/6}}}
{{{x=30}}}
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The angles are 30, 60, 90.  This is a special triangle.  Recognizing this is important.


This special triangle is the result of cutting an equilateral triangle in half along one of the altitudes.  Using another factor for the three sides,  if the longest side is h, this is a hypotenuse of this special triangle.  The shortest leg is h/2, and the longest leg L is as {{{L^2+(h/2)^2=h^2}}},
{{{L^2=h^2-h^2/4}}}
{{{L^2=h^2(1-1/4)}}}
{{{L^2=h^2(3/4)}}}
{{{highlight_green(L=h*sqrt(3)/2)}}}


Use all this perimeter information:
{{{highlight(h+h/2+h*sqrt(3)/2=30+10sqrt(3))}}}
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After solving for h, you can then evaluate the length of each side of the triangle:  {{{system(h, h/2, h*sqrt(3)/2)}}}



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Backup a little bit, and take the perimeter red-outlined equation, and solve for h.
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{{{h(1+1/2+sqrt(3))=30+10sqrt(3)}}}
{{{h(3/2+sqrt(3))=30+10sqrt(3)}}}
{{{h=(30+10sqrt(3))/(3/2+sqrt(3))}}}
simplify using conjugate...
several algebra-arithmetic steps...
{{{highlight(h=20sqrt(3)-20)}}}------Important piece of information, but not yet final answer.


Smaller leg is {{{h/2=(1/2)(20sqrt(3)-20)}}}
{{{highlight(highlight(h/2=10sqrt(3)-10))}}}