https://file.aiquickdraw.com/mathbot/file/1725783178596mhra9l03.jpg

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.

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 vertices?

Determine the coordinates of the remaining vertices of a square with a side length of 2 units, given one vertex at the origin and a diagonal along the x-axis. This problem covers key concepts in coordinate geometry, including the properties of a square, diagonal calculation, and rotation formulas. Ideal for high school students (Grades 9-12).

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