Key Concepts and Formulas

• Polar Coordinates: A point $P$ at distance $r$ from the origin, with the line segment $OP$ making an angle $\theta$ with the positive x-axis, has coordinates $(r \cos \theta, r \sin \theta)$.
• Properties of a Square: All sides are equal in length, and all interior angles are 90 degrees.
• Trigonometric Values: $\cos(30^\circ) = \frac{\sqrt{3}}{2}$, $\sin(30^\circ) = \frac{1}{2}$, $\cos(120^\circ) = -\frac{1}{2}$, $\sin(120^\circ) = \frac{\sqrt{3}}{2}$

Step-by-Step Solution

Step 1: Define the Vertices and Angles

Let the square be $OABC$, where $O$ is the origin $(0,0)$. The side length of the square is $s=2$. $OA$ makes an angle of $30^\circ$ with the positive x-axis. Since $OABC$ is a square, $OC$ makes an angle of $30^\circ + 90^\circ = 120^\circ$ with the positive x-axis.

Step 2: Calculate the Coordinates of Vertex A

Vertex $A$ is at a distance of $2$ from the origin and makes an angle of $30^\circ$ with the x-axis. Using polar coordinates: $A = (2 \cos 30^\circ, 2 \sin 30^\circ) = \left(2 \cdot \frac{\sqrt{3}}{2}, 2 \cdot \frac{1}{2}\right) = (\sqrt{3}, 1)$

Step 3: Calculate the Coordinates of Vertex C

Vertex $C$ is at a distance of $2$ from the origin and makes an angle of $120^\circ$ with the x-axis. Using polar coordinates: $C = (2 \cos 120^\circ, 2 \sin 120^\circ) = \left(2 \cdot \left(-\frac{1}{2}\right), 2 \cdot \frac{\sqrt{3}}{2}\right) = (-1, \sqrt{3})$

Step 4: Calculate the Coordinates of Vertex B

Vertex $B$ can be found by recognizing that $\vec{OB} = \vec{OA} + \vec{OC}$. Thus, the coordinates of $B$ are the sum of the coordinates of $A$ and $C$. $B = (\sqrt{3} + (-1), 1 + \sqrt{3}) = (\sqrt{3} - 1, 1 + \sqrt{3})$

Step 5: Find the Sum of the x-coordinates

The x-coordinates of the vertices are: $x_O = 0$ $x_A = \sqrt{3}$ $x_B = \sqrt{3} - 1$ $x_C = -1$

The sum of the x-coordinates is: $0 + \sqrt{3} + (\sqrt{3} - 1) + (-1) = 2\sqrt{3} - 2$

Common Mistakes & Tips

• Angle Calculation: Be careful when calculating the angles for each vertex. Ensure you are adding or subtracting $90^\circ$ correctly based on the orientation of the square.
• Vector Addition: Using vector addition to find the fourth vertex is generally simpler than trying to find the angle it makes with the x-axis.
• Diagram: Drawing a clear diagram can help visualize the problem and avoid errors in angle calculations.

Summary

The problem involves finding the coordinates of the vertices of a square given its side length and the angle one of its sides makes with the x-axis. We used polar coordinates to find the coordinates of two vertices and vector addition to find the coordinates of the fourth. Finally, we summed the x-coordinates of all four vertices. The sum of the x-coordinates of the vertices of the square is $2\sqrt{3} - 2$.

Final Answer

The final answer is \boxed{2\sqrt 3 - 2}, which corresponds to option (B).