6. Given is a rectangle $A B C D$ where one side is $42 \mathrm{~cm}$ long, and the other is 5 times shorter than the first. Calculate the area of the square $E F G H$ whose perimeter is three times smaller than the perimeter of the given rectangle.
6. 

![](https://cdn.mathpix.com/cropped/2024_05_30_9deb5aafe4aea9d071beg-6.jpg?height=223&width=711&top_left_y=1873&top_left_x=381)

Let \( a \) be the length of one side of the rectangle \( A B C D \), i.e., \( a = 42 \, \text{cm} = 420 \, \text{mm} \).

The length of the other side is \( b = 420 : 5 = 84 \, \text{mm} \).

1 POINT

The perimeter of the given rectangle is:

\( o(A B C D) = 2 \cdot (420 + 84) = 2 \cdot 504 = 1008 \, \text{mm} \)

3 POINTS

![](https://cdn.mathpix.com/cropped/2024_05_30_9deb5aafe4aea9d071beg-6.jpg?height=345&width=346&top_left_y=2489&top_left_x=495)

The perimeter of the square \( E F G H \) is 3 times smaller than the perimeter of the rectangle, so:

\( o(E F G H) = 1008 : 3 = 336 \, \text{mm} \)

2 POINTS

The length of the side of the square is \( 336 : 4 = 84 \, \text{mm} \).

1 POINT

The area of the square is \( 84 \cdot 84 = 7056 \, \text{mm}^2 \). 2 POINTS

TOTAL 10 POINTS