Application of pythagoras theorem

Question

A cathedral spire $200$ ft high is $250$ ft away from a church steple which is $150$ ft high. At the same instant, two crows, one from each, fly of at the same speed heading toward some grain that is on a level, straight road connecting two towers. The crows reach the grain at the same instant. How far along that road is the grain from the cathedral?

Answer 1

Hint: To each triangle you can apply Pythagoras, and then you can go ahead and solve for $x$ and $y$.

Answer 2

Draw a picture. Let $p$ be the distance of the grain from the cathedral, which is a calculus cathedral, namely a line going straight up. Let $q$ be the distance of the grain from the church. Then $q+p=200$.

By the Pythagorean Theorem, $250^2+p^2=150^2+q^2$. So we have the equations $$q+p=200,\qquad\text{and}\qquad q^2-p^2=250^2-150^2=(100)(400).$$ This can be solved pretty quickly. Hint: It is easy to find $q-p$.