Question

Find the area of the square shown in the figure whose vertices are at $(0,0)$, $(1,1)$, $(2,0)$ and $(1,-1)$. 
 

Hint:

For a square rotated in coordinate plane, either compute side using distance formula or use diagonal formula $A=\frac{d^2}{2}$.

Answer 1

Correct Answer: 2

Step 1: Identify the vertices.
From the figure, the four vertices are:
\[ A(0,0), \quad B(1,1), \quad C(2,0), \quad D(1,-1). \]
Step 2: Find the length of one side.
Let us find the distance between $A(0,0)$ and $B(1,1)$.
Using distance formula:
\[ AB = \sqrt{(1-0)^2 + (1-0)^2} \] \[ AB = \sqrt{1 + 1} \] \[ AB = \sqrt{2}. \]
Step 3: Verify adjacent side.
Now check distance between $B(1,1)$ and $C(2,0)$.
\[ BC = \sqrt{(2-1)^2 + (0-1)^2} \] \[ BC = \sqrt{1 + 1} \] \[ BC = \sqrt{2}. \] Thus all sides are equal. Hence it is a square with side $\sqrt{2}$.
Step 4: Compute area.
Area of square is:
\[ \text{Area} = (\text{side})^2 \] \[ = (\sqrt{2})^2 \] \[ = 2. \]
Alternative Method (Using Diagonal).
Diagonal is from $(0,0)$ to $(2,0)$ which equals 2.
Area of square using diagonal formula:
\[ \text{Area} = \frac{(\text{diagonal})^2}{2} \] \[ = \frac{2^2}{2} = \frac{4}{2} = 2. \]
Final Answer:
\[ \boxed{2}. \]