To solve this problem, we need to determine the coordinates of the three remaining vertices of a square with a side length of $2$ units, given that one vertex is at $(0, 0)$ and one of its diagonals is along the x-axis.

Explanation:

1. Square Properties:

   • A square has four sides of equal length and diagonals that bisect each other at right angles (90 degrees).
   • Given that one diagonal is along the x-axis, the center of the square will lie on the x-axis, and the diagonals will be perpendicular.

2. Diagonal Length:

   • The length of a diagonal of a square with a side length $s = 2$ is given by: $\text{Diagonal length} = s\sqrt{2} = 2\sqrt{2}$

3. Finding the Center:

   • Since one vertex is at $(0, 0)$, and the diagonal is along the x-axis, the center of the square will lie at the midpoint of the diagonal. Thus, the center of the square is at $(s\sqrt{2}/2, 0) = (\sqrt{2}, 0)$.

4. Determining the Vertices:

   • The vertices of the square can be found by rotating around the center $(\sqrt{2}, 0)$ by 45 degrees (since the diagonals are perpendicular bisectors).

   Using the rotation formula around the center: $(x', y') = (x_c + (x - x_c) \cos \theta - (y - y_c) \sin \theta, \; y_c + (x - x_c) \sin \theta + (y - y_c) \cos \theta)$ For a 45-degree rotation ($\theta = 45^\circ$):

   • First vertex: Rotating $(0, 0)$ by 45 degrees around $(\sqrt{2}, 0)$ gives $(\sqrt{2}, \sqrt{2})$.
   • Second vertex: Rotating $(2\sqrt{2}, 0)$ by 45 degrees around $(\sqrt{2}, 0)$ gives $(\sqrt{2}, -\sqrt{2})$.
   • The third vertex is already given by the end of the diagonal at $(2\sqrt{2}, 0)$.

5. Final Coordinates: The coordinates of the three other vertices are:

   • $(\sqrt{2}, \sqrt{2})$
   • $(\sqrt{2}, -\sqrt{2})$
   • $(2\sqrt{2}, 0)$

These match the answer provided in the image.

Would you like further details or have any questions?

Here are some related questions for further exploration:

1. What is the equation of a circle circumscribed around this square?
2. How do we find the coordinates of a point after rotating it about another point?
3. How can we determine the coordinates of the vertices of a rectangle instead of a square given one diagonal?
4. What are the properties of a square's diagonals in coordinate geometry?
5. How would the answer change if the side length of the square were different?

Tip: When dealing with coordinate geometry problems involving rotations, always remember to use the rotation formulas or visualize the problem geometrically on a graph.