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.