Problem 4. A water reed whose tip is $20 \mathrm{~cm}$ above the surface of a mountain lake, under the influence of the wind, the water reached and disappeared at a point $60 \mathrm{~cm}$ away from where it originally stood. Assuming the reed is stiff enough to remain straight during the movement, determine the depth of the lake.
Solution. We construct a circle with its center at the bottom of the pole and radius $l=\overline{O D}=\overline{O A}$ - the length of the pole (draw a diagram). Note that triangle $\triangle A D E$ is a right triangle, so triangle $\triangle A B D$ is similar to $\triangle D B E$, which gives us $\frac{\overline{A B}}{\overline{B D}}=\frac{\overline{B D}}{\overline{B E}}$, meaning that $\overline{B E}=\frac{\overline{B D}^{2}}{\overline{A B}}$. On the other hand, $\overline{B E}=2 x+20$, so $x=80 \mathrm{~cm}$.

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