5. The image shows a rectangle with a square cut out. The area of the resulting shape is 369 $\mathrm{cm}^{2}$, and the lengths of the shortened sides are $21 \mathrm{~cm}$ and $9 \mathrm{~cm}$. What is the area of the cut-out square?

![](https://cdn.mathpix.com/cropped/2024_06_03_c764e3a28b36bcb00b07g-1.jpg?height=422&width=742&top_left_y=2025&top_left_x=677)

Each problem is scored out of 10 points.

The use of a pocket calculator or any reference materials is not allowed.
5. Let the side of the square be denoted by $a$, and the area of the shape by $P$.

We divide the obtained shape into three parts whose areas are $P_{1}, P_{2}$, and $P_{3}$.

![](https://cdn.mathpix.com/cropped/2024_05_30_c9769736c1775f57880fg-4.jpg?height=531&width=945&top_left_y=277&top_left_x=290)[^0]


[^0]:    The area $P_{1}$ is $P_{1}=21 \mathrm{~cm} \cdot 9 \mathrm{~cm}=189 \mathrm{~cm}^{2}$.

    2 POINTS

    Then, $P_{2}+P_{3}=P-P_{1}=369 \mathrm{~cm}^{2}-189 \mathrm{~cm}^{2}=180 \mathrm{~cm}^{2}$. 2 POINTS

    It holds that:

    $P_{2}=21 a$,

    $P_{3}=9 a$, so

    $P_{2}+P_{3}=21 a+9 a=30 a$.

    2 POINTS

    From $30 a=180$ we calculate

    $a=180: 30=6 \mathrm{~cm}$.

    2 POINTS

    The area of the square is $36 \mathrm{~cm}^{2}$.

    2 POINTS

    TOTAL 10 POINTS