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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.

A square of side 2 units has one vertex at (0, 0) and one diagonal along the x-axis. What are the coordinates of the three other vertices?

This problem involves finding the coordinates of the vertices of a square with side length 2 units, given that one vertex is at (0, 0) and the diagonal lies along the x-axis. Learn how to apply the diagonal formula and rotational symmetry to solve it.

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