The area of a square field is `(1)/(2)` hectare. Find the length of its diagonal in metres.
[Hint. 1 hectare = `10000 m^(2)`]

Answer

To solve the problem step by step, we will follow these instructions:

### Step 1: Convert the area from hectares to square meters.
Given that the area of the square field is \( \frac{1}{2} \) hectare, we need to convert this to square meters using the conversion factor \( 1 \text{ hectare} = 10,000 \text{ m}^2 \).

\[ \text{Area in m}^2 = \frac{1}{2} \times 10,000 = 5,000 \text{ m}^2 \]

### Step 2: Use the area to find the side length of the square.
The formula for the area of a square is given by:

\[ \text{Area} = \text{side}^2 \]

We can set up the equation:

\[ \text{side}^2 = 5,000 \]

To find the side length, we take the square root of both sides:

\[ \text{side} = \sqrt{5,000} \]

### Step 3: Simplify the square root.
We can simplify \( \sqrt{5,000} \):

\[ \sqrt{5,000} = \sqrt{5 \times 1,000} = \sqrt{5} \times \sqrt{1,000} = \sqrt{5} \times \sqrt{100 \times 10} = \sqrt{5} \times 10 \times \sqrt{10} = 10\sqrt{50} \]

Since \( 50 = 25 \times 2 \), we can further simplify:

\[ 10\sqrt{50} = 10 \times 5 \times \sqrt{2} = 50\sqrt{2} \text{ m} \]

### Step 4: Calculate the diagonal of the square.
The formula for the diagonal \( d \) of a square in terms of its side length \( s \) is:

\[ d = s \times \sqrt{2} \]

Substituting the value of the side we found:

\[ d = 50\sqrt{2} \times \sqrt{2} = 50 \times 2 = 100 \text{ m} \]

### Conclusion
The length of the diagonal of the square field is \( 100 \text{ m} \).

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