4. In rectangle $A B C D$ with sides $A B=10 \mathrm{~cm}$ and $B C=5 \mathrm{~cm}$, a semicircle with diameter $A B$ is inscribed. In what ratio does the semicircle divide the diagonal of the rectangle?
Solution. Let $E$ be the intersection point of the diagonal $A C$ with the semicircle $k$ (diagram on the right). Then $\measuredangle A E B=$ $90^{\circ}$, as an angle over the diameter $A B$. The right triangles $A B E$ and $B C E$ are similar, as they have equal acute angles. Therefore,

$$
A E: E C=A B: B C=10: 5=2: 1
$$