In the following figure, the square $A B C D$ has been divided into three rectangles of the same area.

![](https://cdn.mathpix.com/cropped/2024_05_01_d1e5bbfbdffc7558a598g-12.jpg?height=423&width=452&top_left_y=822&top_left_x=678)

If the length of the segment $B M$ is equal to 4, calculate the area of the square $A B C D$.

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Solution

In the following figure, the area of rectangle $M N P C$ is $|M N| \times|N P|$, while the area of rectangle $N Q D P$ is $|N Q| \times|N P|$.

![](https://cdn.mathpix.com/cropped/2024_05_01_d1e5bbfbdffc7558a598g-35.jpg?height=587&width=603&top_left_y=449&top_left_x=608)

For the areas of these rectangles to coincide, it is necessary that $|M N|$ and $|N Q|$ be equal. Let's call this common value $a$, that is, $a=|M N|=|N Q|$. Thus, we have $|M Q|=|M N|+|N Q|=2 a$. Therefore, the area of rectangle $A B M Q$ is equal to $|B M| \times|M Q|=4 \times(2 a)=8 a$. The area of rectangle $M N P C$ is equal to $|M N| \times|N P|=a \times|N P|$. Thus, for the areas of rectangles $M N P C$ and $N Q D P$ to also be equal to $8 a$, it is necessary that $N P=8$.

Therefore, the measure of side $B C$ of the square coincides with $|B M|+|M C|=$ $|B M|+|N P|=4+8=12$. The area of square $A B C D$ is, therefore, $12^{2}=144$.