The problem states that a square has a side length of 2 units, with one vertex at $(0,0)$ and one diagonal along the x-axis. We need to find the coordinates of the other three vertices.

Step-by-Step Solution:

1. First Vertex: One vertex is already given as $A(0, 0)$.

2. Diagonal Properties:

   • A square's diagonal is $\sqrt{2}$ times the side length.
   • For this square, the diagonal length will be: $\text{Diagonal length} = 2 \times \sqrt{2} = 2\sqrt{2}$
   • Since the diagonal is aligned along the x-axis, we know the other vertex on the diagonal must be at $(2\sqrt{2}, 0)$.

3. Midpoint of the Diagonal:

   • The midpoint of the diagonal lies at the center of the square. For a diagonal along the x-axis, the midpoint is: $\text{Midpoint} = \left( \frac{0 + 2\sqrt{2}}{2}, \frac{0 + 0}{2} \right) = (\sqrt{2}, 0)$ This is the center of the square.

4. Rotation to Find Remaining Vertices:

   • Since the square is symmetric about its center, the remaining two vertices can be obtained by rotating points by 90° around the center $(\sqrt{2}, 0)$.
   • The remaining vertices will be equidistant from the center, lying at distances of $1$ unit from the center, both above and below the x-axis.

Coordinates of the vertices:

• First vertex: $(0,0)$
• Second vertex: $(2\sqrt{2}, 0)$
• Third vertex (rotated): $(\sqrt{2}, \sqrt{2})$
• Fourth vertex (rotated): $(\sqrt{2}, -\sqrt{2})$

Final Answer:

The coordinates of the three other vertices are:

1. $(2\sqrt{2}, 0)$
2. $(\sqrt{2}, \sqrt{2})$
3. $(\sqrt{2}, -\sqrt{2})$

Would you like more details on this approach or clarification? Here are some related questions:

1. How can we generalize this solution for squares of different sizes?
2. Why is the diagonal length of a square always $\sqrt{2}$ times the side length?
3. What are the steps to rotate points around a given center?
4. How would the problem change if the diagonal were along a different axis?
5. What if the vertex was not at $(0,0)$? How would the solution change?

Tip: When dealing with squares, always use the symmetry properties of the figure to simplify finding vertices, sides, and diagonals.