jmc

We might try sketching a diagram: Since $BC$ is a diameter of the circle, that makes $\angle BDC$ a right angle. That means that $BD$ is an altitude of $△ABC.$ Then, we use the area formula to find $150=\frac{1}{2}\cdot AC\cdot BD,$ where $AC=25.$ Solving, we have $BD=\boxed{12}.$