Side of the square field = 21 mPerimeter of the square field = (4 × 21) m= 84 mLet the length and breadth of the rectangular field be 4x and 3x, respectively.Perimeter of the rectangular field = Perimeter of the square fieldTherefore, \(14x = 84 \)\( \Rightarrow x = \frac {84} {14} = 6 \)Therefore, \(Length \; of\; the\; rectangular\; field = \left ( 4 \times 6 \right ) m = 24 \; m \)\( Breadth \; of\; the\; rectangular\; field = \left ( 3 \times 6 \right )m = 18 \; m \)

Solution:

Let's start by finding the perimeter of the square field. The perimeter of a square is given by the formula:

Perimeter = 4 * side

Since each side of the square field measures 21cm, the perimeter of the square field is:

Perimeter of square = 4 * 21cm = 84cm

Now, let's find the dimensions of the rectangular field adjacent to the square field. The sides of the rectangular field are in the ratio 4:3, which means that one side is 4x and the other side is 3x, where x is a common factor.

Let's assume the common factor x to be 1. Therefore, one side of the rectangular field is 4 * 1 = 4cm and the other side is 3 * 1 = 3cm.

The perimeter of a rectangle is given by the formula:

Perimeter = 2 * (length + width)

Since the sides of the rectangular field are 4cm and 3cm, the perimeter of the rectangular field is:

Perimeter of rectangle = 2 * (4cm + 3cm) = 2 * 7cm = 14cm

Since the perimeters of both fields are equal, we can equate the two perimeter values:

84cm = 14cm

Simplifying the equation, we get:

84cm = 14cm

Canceling out the common factor of 14cm on both sides, we get:

6 = 1

This equation is not true, which means that our assumption of x = 1 is incorrect. We need to find the correct value of x.

Let's assume the common factor x to be 2. Therefore, one side of the rectangular field is 4 * 2 = 8cm and the other side is 3 * 2 = 6cm.

The perimeter of the rectangular field is:

Perimeter of rectangle = 2 * (8cm + 6cm) = 2 * 14cm = 28cm

Now, let's compare the perimeters of the square field and the rectangular field:

Perimeter of square = 84cm
Perimeter of rectangle = 28cm

Since the perimeters are not equal, our assumption of x = 2 is incorrect. We need to try a different value for x.

Let's assume the common factor x to be 3. Therefore, one side of the rectangular field is 4 * 3 = 12cm and the other side is 3 * 3 = 9cm.

The perimeter of the rectangular field is:

Perimeter of rectangle = 2 * (12cm + 9cm) = 2 * 21cm = 42cm

Now, let's compare the perimeters of the square field and the rectangular field:

Perimeter of square = 84cm
Perimeter of rectangle = 42cm

Since the perimeters are equal, we have found the correct dimensions of the rectangular field.

Dimensions of the Fields:
- Square field: Each side measures 21cm
- Rectangular field: One side measures 12cm and the other side measures 9cm