Given that a square has one vertex at
(0,0), a diagonal along the x-axis, and a side length of 2 units, we can find the coordinates of the other three vertices using basic geometry.

Given that a square has:
- One vertex at \((0, 0)\),
- A diagonal along the x-axis,
- A side length of 2 units,

we want to find the coordinates of the other three vertices.

### Step-by-Step Solution

1. **Identify the diagonal of the square:**

   Since the diagonal is along the x-axis, we can infer that:
   - One endpoint of the diagonal is at \((0, 0)\).
   - The length of the diagonal of a square with side length \(s = 2\) is:
   
   \[
   \text{Diagonal} = s \sqrt{2} = 2\sqrt{2}
   \]

   Therefore, the other endpoint of the diagonal must be at \((2\sqrt{2}, 0)\) along the x-axis.

2. **Find the coordinates of the other two vertices:**

   The square's vertices must be perpendicular to each other, and the remaining two vertices will be located symmetrically on either side of the x-axis.

   Let's denote the other two vertices as \((x_1, y_1)\) and \((x_2, y_2)\).

   Given the diagonal lies along the x-axis, the line connecting the other two vertices is perpendicular to the diagonal.

   The center of the square, which is the midpoint of the diagonal, will be at:

   \[
   \left( \frac{0 + 2\sqrt{2}}{2}, \frac{0 + 0}{2} \right) = (\sqrt{2}, 0)
   \]

   The distance from the center to each of these vertices is:

   \[
   \frac{s}{2} = \frac{2}{2} = 1 \text{ unit}
   \]

   The remaining vertices will be located 1 unit above and below the x-axis, along the line perpendicular to the x-axis through \((\sqrt{2}, 0)\).

3. **Determine the exact coordinates:**

   Therefore, the coordinates of the remaining vertices are:

   \[
   (\sqrt{2}, 1) \quad \text{and} \quad (\sqrt{2}, -1)
   \]

### Conclusion

The coordinates of the four vertices of the square are:

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

Would you like more details or have any questions? Here are some related topics to explore:

1. What is the equation of the line passing through two points?
2. How do you calculate the diagonal length of a square?
3. What are the properties of perpendicular bisectors in geometry?
4. How do you find the midpoint of a line segment?
5. How can we find the vertices of a rectangle given one side and length?

**Tip:** Knowing the properties of symmetry and perpendicular lines can help solve many geometric problems more efficiently.

Given that a square has one vertex at (0,0), a diagonal along the x-axis, and a side length of 2 units, we can find the coordinates of the other three vertices using basic geometry.

Discover how to calculate the coordinates of the vertices of a square with one vertex at (0,0), a diagonal along the x-axis, and a side length of 2 units. Using basic geometry and symmetry principles, this solution includes key concepts such as the diagonal formula and midpoint calculation. Suitable for students in Grades 9-10.

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