## 55. Math Puzzle 12/69

On the banks of a river, two palm trees stand opposite each other. One is 30 cubits tall, the other 20 cubits; the distance between them is 50 cubits.

At the top of each palm tree sits a bird. Both notice a fish that has swum to the surface of the water between the two palm trees and dive for it at the same time, reaching the fish simultaneously.

At what distance from the location of the taller palm tree did the fish appear?
From the drawing, it is evident that according to the Pythagorean theorem:

$$
\overline{A B}^{2}=30^{2}+x^{2} \quad \text { and } \quad \overline{A C}^{2}=20^{2}+(50-x)^{2}
$$

Since for the problem $\overline{A B}=\overline{A C}$ is set, we get $30^{2}+x^{2}=20^{2}+(50-x)^{2}$ with the solution $x=20$. The fish appeared 20 ells away from the larger palm tree.