The diagonal of a square field is 25 metres. The area of the field is-

Options

A. 625 sq. metres
B. `312cdot5` sq. metres
C. `156cdot25` sq. metres
D. `(625)/sqrt2` sq. metres

Answer

To find the area of a square field when the diagonal is given, we can follow these steps:

### Step 1: Understand the relationship between the diagonal and the side of the square
The diagonal \(d\) of a square can be related to its side length \(a\) using the Pythagorean theorem. In a square, the diagonal forms a right triangle with two sides of the square. Therefore, we can express the relationship as:
\[ d = a\sqrt{2} \]

### Step 2: Substitute the given diagonal length
We know that the diagonal \(d\) is 25 meters. We can substitute this value into the equation:
\[ 25 = a\sqrt{2} \]

### Step 3: Solve for the side length \(a\)
To find \(a\), we can rearrange the equation:
\[ a = \frac{25}{\sqrt{2}} \]
To simplify this, we can multiply the numerator and denominator by \(\sqrt{2}\):
\[ a = \frac{25\sqrt{2}}{2} \]

### Step 4: Calculate the area of the square
The area \(A\) of a square is given by the formula:
\[ A = a^2 \]
Now, substituting the value of \(a\):
\[ A = \left(\frac{25\sqrt{2}}{2}\right)^2 \]
Calculating this gives:
\[ A = \frac{625 \cdot 2}{4} = \frac{1250}{4} = 312.5 \text{ square meters} \]

### Final Answer
The area of the square field is \(312.5\) square meters.

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