To determine the area of a square in terms of its diagonal [tex]\( x \)[/tex], we can follow these detailed steps:
1. Relationship Between Diagonal and Side Length:
The diagonal [tex]\( d \)[/tex] of a square with side length [tex]\( s \)[/tex] can be derived using the Pythagorean theorem in the right triangle formed by two sides and the diagonal:
[tex]\[
d = s\sqrt{2}
\][/tex]
Given that the diagonal is [tex]\( x \)[/tex] units, we have:
[tex]\[
x = s\sqrt{2}
\][/tex]
2. Solving for Side Length:
We can isolate the side length [tex]\( s \)[/tex] by dividing both sides of the equation by [tex]\( \sqrt{2} \)[/tex]:
[tex]\[
s = \frac{x}{\sqrt{2}}
\][/tex]
3. Calculating the Area:
The area [tex]\( A \)[/tex] of a square is given by the square of its side length:
[tex]\[
A = s^2
\][/tex]
Substituting [tex]\( s = \frac{x}{\sqrt{2}} \)[/tex] into the area formula:
[tex]\[
A = \left( \frac{x}{\sqrt{2}} \right)^2
\][/tex]
4. Simplifying the Expression:
We simplify the squared term:
[tex]\[
A = \frac{x^2}{(\sqrt{2})^2} = \frac{x^2}{2}
\][/tex]
Therefore, the area of the square in terms of the diagonal [tex]\( x \)[/tex] is:
[tex]\[
\frac{x^2}{2} \text{ square units}
\][/tex]
The correct answer is [tex]\(\boxed{\frac{1}{2} x^2}\)[/tex] square units.
