If the diagonal of a square is 10 cm, then what is the area (in `cm^(2)`) of the square?

Options

A. 100
B. `50sqrt""2`
C. 50
D. `100sqrt""2`

Answer

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

### Step 1: Understand the relationship between the side of the square and its diagonal.
The diagonal \( d \) of a square is related to its side length \( a \) by the formula:
\[ d = a \sqrt{2} \]

### Step 2: Substitute the given diagonal into the formula.
Given that the diagonal \( d = 10 \) cm, we can set up the equation:
\[ 10 = a \sqrt{2} \]

### Step 3: Solve for the side length \( a \).
To find \( a \), we rearrange the equation:
\[ a = \frac{10}{\sqrt{2}} \]

### Step 4: Simplify the expression for \( a \).
To simplify \( a \), we can multiply the numerator and the denominator by \( \sqrt{2} \):
\[ a = \frac{10 \sqrt{2}}{2} = 5\sqrt{2} \text{ cm} \]

### Step 5: Calculate the area of the square.
The area \( A \) of the square is given by:
\[ A = a^2 \]
Substituting the value of \( a \):
\[ A = (5\sqrt{2})^2 \]

### Step 6: Perform the squaring operation.
Calculating \( (5\sqrt{2})^2 \):
\[ A = 25 \times 2 = 50 \text{ cm}^2 \]

### Final Answer:
The area of the square is \( 50 \text{ cm}^2 \).

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