Step 1: Understanding the Question
The question asks for the area of a square whose vertices are given as points in a Cartesian coordinate system. Step 2: Key Formula or Approach
There are two common methods to find the area of a square from its vertex coordinates:
1. Side Length Method: Calculate the length of one side using the distance formula, and then square it. The distance between two points $(x_1, y_1)$ and $(x_2, y_2)$ is $d = \sqrt{(x_2-x_1)^2 + (y_2-y_1)^2}$. The area is $d^2$.
2. Diagonal Length Method: Calculate the length of a diagonal using the distance formula. The area is then given by $\frac{(\text{diagonal})^2}{2}$. Step 3: Detailed Explanation (Using Side Length Method)
Let's identify the vertices as: $A(0,0)$, $B(1,1)$, $C(2,0)$, and $D(1,-1)$.
We will calculate the length of the side AB by finding the distance between points A and B.
\[
\text{Side length} = \sqrt{(1-0)^2 + (1-0)^2}
\]
\[
= \sqrt{1^2 + 1^2}
\]
\[
= \sqrt{1 + 1} = \sqrt{2}
\]
The area of a square is the square of its side length.
\[
\text{Area} = (\text{side length})^2 = (\sqrt{2})^2 = 2
\]
Alternative Method (Using Diagonal Length)
The diagonals of this square connect A(0,0) to C(2,0) and B(1,1) to D(1,-1).
Let's find the length of the diagonal AC.
\[
\text{Diagonal length} = \sqrt{(2-0)^2 + (0-0)^2}
\]
\[
= \sqrt{2^2} = 2
\]
Now, we use the formula for the area of a square based on its diagonal.
\[
\text{Area} = \frac{(\text{diagonal})^2}{2} = \frac{2^2}{2} = \frac{4}{2} = 2
\]
Both methods yield the same result. Step 4: Final Answer
The area of the square is 2 square units.
\[
\boxed{2}
\]
