The area of the square-shaped field is given as $121 \, \text{m}^2$. To find the area of the rectangular road along the diagonal of the square, we first need to find the length of the diagonal and then use the breadth of the road (which is 1 meter) to calculate the area.

Step 1: Find the side length of the square

The area $A$ of a square is related to the side length $s$ by the formula: $A = s^2$ Given that $A = 121 \, \text{m}^2$, we can solve for $s$: $s = \sqrt{121} = 11 \, \text{m}$ So, the side length of the square is $11 \, \text{m}$.

Step 2: Find the diagonal of the square

The diagonal $d$ of a square is related to the side length by the formula: $d = s\sqrt{2}$ Substituting $s = 11 \, \text{m}$: $d = 11\sqrt{2} \, \text{m} \approx 11 \times 1.414 = 15.556 \, \text{m}$ So, the length of the diagonal is approximately $15.556 \, \text{m}$.

Step 3: Calculate the area of the rectangular road

The area $A_{\text{road}}$ of the rectangular road is given by the product of its length (which is the diagonal of the square) and its breadth (which is $1 \, \text{m}$): $A_{\text{road}} = d \times \text{breadth} = 15.556 \times 1 = 15.556 \, \text{m}^2$ Thus, the area of the road is approximately $15.556 \, \text{m}^2$.

Final Answer:

The area of the road is approximately $15.56 \, \text{m}^2$.

Would you like more details or have any questions?

Here are five related questions:

1. How do you calculate the diagonal of a square using side length?
2. What is the relationship between area and side length in a square?
3. How would the area of the road change if the breadth of the road were 2 meters?
4. How is the Pythagorean theorem related to finding the diagonal of a square?
5. If the area of the square field were doubled, what would happen to the area of the road?

Tip: In geometry, knowing basic relationships like the one between a square's diagonal and its side length can simplify solving more complex problems.