Find the length of the shortest altitude of a triangle with sides of lengths $15$, $20$, and $25$.

Answer 1

Per law of sines and cosines, the angles are

36.8698976458...

53.13010235413....

and 90

So one of the heights is 20*sin 36.8698976458... which converges to 12

The other height is 15*sin 53.13010235413... which converges to 12

So 12

Answer 2

What if the sides were 3, 4 and 5?