## Task 3 - 280623

Rolf draws a rectangle. He then reduces the longer side length by $2 \mathrm{~cm}$ and notices:

This results in a second rectangle whose area is $8 \mathrm{~cm}^{2}$ smaller than the area of the first rectangle.

Furthermore, he increases both side lengths of the first rectangle by $1 \mathrm{~cm}$ each and notices:

This results in a third rectangle whose area is $13 \mathrm{~cm}^{2}$ larger than the area of the first rectangle.

Show that Rolf's observations alone allow the determination of the two side lengths of the first rectangle!

State these side lengths!
Reducing the larger side length of the first rectangle can be achieved by cutting off a sub-rectangle whose one side length is $2 \mathrm{~cm}$ and whose area is $8 \mathrm{~cm}^{2}$.

Since $8: 2=4$, the other side length of this sub-rectangle is $4 \mathrm{~cm}$. This is also the smaller side length of the first rectangle; thus, it is uniquely determined.

Increasing both side lengths of the first rectangle can be achieved by adding two sub-rectangles, one with a side length of $1 \mathrm{~cm}$ and (since $4+1=5$) $5 \mathrm{~cm}$, thus having an area of $5 \mathrm{~cm}^{2}$, and the other with a side length of $1 \mathrm{~cm}$ and (since $13-5=8$) an area of $8 \mathrm{~cm}^{2}$. Since $8: 1=8$, the other side length of this sub-rectangle is $8 \mathrm{~cm}$. This is also the larger side length of the first rectangle; thus, it is uniquely determined.

![](https://cdn.mathpix.com/cropped/2024_06_06_b85a894aa8dbf722a7b3g-0266.jpg?height=315&width=536&top_left_y=228&top_left_x=266)

In the figure, each colored area has an area of 13 $\mathrm{cm}^{2}$.

Thus, it can be uniquely determined: The side lengths of the first rectangle are $4 \mathrm{~cm}$ and $8 \mathrm{~cm}$.