Question

the diagonal of a square is ( x ) units. what is the area of the square in terms of ( x )? ( 2x ) square units ( \frac{1}{2}x ) square units ( \frac{1}{2}x^{2} ) square units ( x^{2} ) square units

Explanation:

Step1: Let the side of the square be \(a\)

By Pythagorean theorem, \(a^{2}+a^{2}=x^{2}\) (diagonal relation in square).

Step2: Simplify the equation

\(2a^{2}=x^{2}\), so \(a^{2}=\frac{x^{2}}{2}\).

Step3: Recall the area formula of square

Area of square \(A = a^{2}\). Substituting \(a^{2}=\frac{x^{2}}{2}\), we get \(A=\frac{1}{2}x^{2}\).

Answer:

\(\frac{1}{2}x^{2}\) square units