In rectangle $K L M N$, the distance from the intersection of the diagonals to line $K L$ is $2 \mathrm{~cm}$ less than its distance from line $L M$. The perimeter of the rectangle is $56 \mathrm{~cm}$. What is the area of rectangle $K L M N$?

(L. Hozová)
The intersection of the diagonals in rectangle $K L M N$ is denoted as $S$. If the distance from point $S$ to side $K L$ is $2 \mathrm{~cm}$ less than the distance to side $L M$, this means that side $K L$ is $2+2=4(\mathrm{~cm})$ longer than side $L M$.

![](https://cdn.mathpix.com/cropped/2024_04_17_a7af8483648468ce1827g-2.jpg?height=691&width=922&top_left_y=611&top_left_x=567)

If we shorten the length of the rectangle (i.e., sides $K L$ and $M N$) by $4 \mathrm{~cm}$, we obtain a square whose side is the same as the width of the original rectangle. This square has a perimeter of $56-2 \cdot 4=48(\mathrm{~cm})$.

![](https://cdn.mathpix.com/cropped/2024_04_17_a7af8483648468ce1827g-2.jpg?height=691&width=914&top_left_y=1536&top_left_x=571)

The side of the square, and thus the sides $L M$ and $N K$ of the rectangle, measure $48: 4=12(\mathrm{~cm})$. This means that the sides $K L$ and $M N$ measure $12+4=16(\mathrm{~cm})$. The area of the rectangle is therefore $S=16 \cdot 12=192\left(\mathrm{~cm}^{2}\right)$.

Evaluation: 2 points for determining the relationship between sides $K L$ and $L M$; 2 points for determining the lengths of sides $K L$ and $L M$ using the perimeter; 2 points for determining the area of the rectangle.