Partial Solution

I don't get this question..."triangle that is perpendicular to the longest side"??

franzliszt wrote: I don't get this question..."triangle that is perpendicular to the longest side"?? read the words that are right before it.

Still don't get it. "What is the length of the altitude (height) of the triangle that is perpendicular to the longest side?" How to make a triangle perpendicular to the longest side?

franzliszt wrote: Still don't get it. "What is the length of the altitude (height) of the triangle that is perpendicular to the longest side?" How to make a triangle perpendicular to the longest side? It means "What is the length of the altitude that is perpendicular to the longest side." Like in a 30-40-50 triangle that height is $24$

Oh then this is bashy? You do LoS bash to find the sides. Then use Heron's. Then set and equation to find the altitude.

A non-bashing sol., it is a 15-75-90 triangle, so we have \[\frac{1}{2}(12\sin 15)(12\cos 15)=\frac{1}{2}(12)(h) \implies h=12 \sin 15 \cos 15=6\sin 30=\boxed{3}\]