To solve for the area of a square given its diagonal, let's follow these steps:
1. Understand the relation between the side and the diagonal:
- In a square, if the length of the diagonal is [tex]\( x \)[/tex] units, we know that each side of the square is [tex]\( s \)[/tex] units.
- The relationship between the side length [tex]\( s \)[/tex] and the diagonal [tex]\( x \)[/tex] in a square is derived from the Pythagorean theorem. For a square with side length [tex]\( s \)[/tex], the diagonal forms a right triangle with the two sides of the square.
- Therefore, [tex]\( x^2 = s^2 + s^2 \)[/tex].
2. Apply the Pythagorean theorem:
- This simplifies to [tex]\( x^2 = 2s^2 \)[/tex].
- Solving for [tex]\( s^2 \)[/tex], we get: [tex]\( s^2 = \frac{x^2}{2} \)[/tex].
3. Calculate the area of the square:
- The area of a square is given by the side length squared. So, [tex]\( \text{Area} = s^2 \)[/tex].
- Substituting [tex]\( s^2 \)[/tex] from the above equation, we have [tex]\( \text{Area} = \frac{x^2}{2} \)[/tex].
Therefore, the area of the square in terms of the diagonal [tex]\( x \)[/tex] is [tex]\(\frac{1}{2} x^2 \)[/tex] square units.
Hence, the correct answer is [tex]\(\frac{1}{2} x^2\)[/tex] square units.
