Question 429889
One of the other tutors mis-applied Heron's formula. Instead of {{{A = sqrt((p-a)(p-b)(p-c))}}}, it is {{{A = sqrt(p(p-a)(p-b)(p-c))}}}. Heron's formula works, but it is somewhat tedious. Also, we could use the standard {{{A = bh/2}}} but that is boring, and there are faster solutions.


We can let the side length of the triangle be x. We can write the area in terms of two of the side lengths and the angle in between, i.e.


{{{x^2*sin(60)/2 = 12sqrt(3)}}}


{{{x^2*sin(60) = 24sqrt(3)}}}. Since {{{sin(60) = sqrt(3)/2}}},


{{{x^2*sqrt(3)/2 = 24sqrt(3)}}}


{{{x^2 = 48}}}


{{{x = sqrt(48)}}}, approximately 6.928.