The height of two triangles My textbook says the height of the following triangles ($BC$) is $37.5$ 

Because
$$
\widehat{B_1}=\widehat{B_2}=30  \Rightarrow BD=50  \Rightarrow DC=25  \Rightarrow AC=75
$$
and since in right triangles the side opposite the 30 degrees angle is half the length of the hypotenuse, the height is $37.5$.
But in my approach the answer is $25 \sqrt{3}$ because
$$
\tan{30}=\frac{1}{\sqrt{3}}=\frac{BC}{50+DC}, \quad \tan{60}=\sqrt{3}=\frac{BC}{DC}\to DC=\frac{BC}{\sqrt{3}}\\
\to \frac{1}{\sqrt{3}}=\frac{BC}{50+\frac{BC}{\sqrt{3}}} \to BC=25\sqrt{3} \approx 43.3
$$
Am I right?

I think you are. I used a different approach, but got your same answer. Your textbook says "in right triangles the side opposite the 30 degrees is half the length of the hypotenuse", but $AC$ is not the hypotenuse.
Here is my work:
$AC=75$
using the laws of 30-60-90 triangles,
$75=x\sqrt{3}$
$x=\frac{75}{\sqrt{3}}$
which simplifies to
$x=25\sqrt{3}$
