Question 407148
Use the Law of Sines:


{{{a/(sin(30)) = b/(sin(60)) = c/(sin(90))}}}


Substitute {{{sin(30) = 1/2}}}, {{{sin(60) = sqrt(3)/2}}}, {{{sin(90) = 1}}} to get


{{{a/(1/2) = b/(sqrt(3)/2) = c}}}


{{{2a = 2b/sqrt(3) = c}}}. Here, we get c = 2a, and {{{b = a*sqrt(3)}}} as desired.


Interesting fact: The law of sines can be extended and it actually says
{{{a/(sin(alpha)) = b/(sin(beta)) = c/(sin(gamma)) = 2R}}}
where {{{alpha}}}, {{{beta}}}, {{{gamma}}} are the angles opposite a,b,c and R is the circumradius.