Calculate dimensions of the rectangle

Given the ratio $1: 4$, let the sides be $x$ and $4x$. The perimeter is $2(l+w) = 50$. Substituting the values: $2(x + 4x) = 50 \implies 10x = 50 \implies x = 5$. Thus, sides are $5$ cm and $20$ cm.

$l = 20, w = 5$

Calculate the area of the rectangle

The area of the rectangle is $l \times w = 20 \times 5 = 100$ cm².

$\text{Area} = 100 \text{ cm}^2$

Calculate the side of the square

The square has the same area as the rectangle, so $a^2 = 100$. Taking the square root, $a = 10$ cm.

$a = \sqrt{100} = 10 \text{ cm}$

Calculate the perimeter of the square

The perimeter of a square is $4 \times a$. Therefore, $4 \times 10 = 40$ cm.

$P = 4 \times 10 = 40 \text{ cm}$