3A. Let a circle with diameter $A B$ be given, and let $C$ be a point on the circle different from $A$ and $B$. Let $M$ be the midpoint of the chord $B C$, and the distance from $M$ to $A B$ is $1 \mathrm{~cm}$. If $\measuredangle B A C=30^{\circ}$, calculate the length of the chord $A C$.
Solution. Let $M_{1}$ be the orthogonal projection of $M$ onto $AB$, and $C_{1}$ be the orthogonal projection of $C$ onto $AB$. From $\overline{M M_{1}}=1 \mathrm{~cm}$ and $M M_{1}$ is the midline in $\triangle C_{1} B C$, it follows that $\overline{C C_{1}}=2 \overline{M M_{1}}=2 \mathrm{~cm}$. From $\triangle A C_{1} C$ being a right triangle and $\measuredangle C_{1} A C=30^{\circ}$, it follows that the hypotenuse $\overline{A C}=2 \overline{C C_{1}}=4 \mathrm{~cm}$.

4AB. The rectangle $ABCD$ is divided into 9 smaller rectangles, such that the areas of four of them are $8, 10, 5, 12$ as shown in the diagram. Determine the smallest possible value of the area of rectangle $ABCD$.

![](https://cdn.mathpix.com/cropped/2024_06_05_e56627a2e774dea33268g-05.jpg?height=334&width=387&top_left_y=244&top_left_x=1095)