If the diagonal of a square field is 16 m, what is its area?

Options

A. 132 `m^2`
B. 130 `m^2`
C. 128 `m^2`
D. 126 `m^2`

Answer

To find the area of a square field given that its diagonal is 16 m, we can follow these steps:

### Step 1: Understand the relationship between the diagonal and the sides of the square.
In a square, the diagonal divides the square into two right-angled triangles. According to the Pythagorean theorem, if 'a' is the length of a side of the square, then the relationship between the side and the diagonal (d) is given by:
\[ d = a\sqrt{2} \]

### Step 2: Substitute the given diagonal length into the equation.
We know the diagonal (d) is 16 m. Therefore, we can set up the equation:
\[ 16 = a\sqrt{2} \]

### Step 3: Solve for 'a'.
To find 'a', we can rearrange the equation:
\[ a = \frac{16}{\sqrt{2}} \]

### Step 4: Simplify the expression for 'a'.
To simplify \(\frac{16}{\sqrt{2}}\), we can multiply the numerator and the denominator by \(\sqrt{2}\):
\[ a = \frac{16\sqrt{2}}{2} = 8\sqrt{2} \]

### Step 5: Calculate the area of the square.
The area (A) of a square is given by the formula:
\[ A = a^2 \]
Substituting the value of 'a':
\[ A = (8\sqrt{2})^2 \]
\[ A = 64 \cdot 2 = 128 \, \text{m}^2 \]

### Final Answer:
The area of the square field is \(128 \, \text{m}^2\).

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