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

Solution Approach:

• Since the square has a diagonal along the x-axis, and one vertex at the origin, let's first calculate the length of the diagonal. The diagonal of a square is given by: $\text{Diagonal} = \text{side length} \times \sqrt{2}$ For a square with a side of 2 units: $\text{Diagonal} = 2 \times \sqrt{2} = 2\sqrt{2}$
• One diagonal lies along the x-axis, starting from $(0, 0)$, which means the other end of this diagonal will be at $(2\sqrt{2}, 0)$.

Now, let's calculate the coordinates of the other two vertices. Since the square has its sides perpendicular to each other, we need to rotate the diagonal by $90^\circ$.

1. One vertex is at $(0, 0)$.
2. The second vertex is directly along the x-axis at $(2\sqrt{2}, 0)$.
3. The other two vertices are symmetric to these two along the diagonal, and their positions can be derived using rotations.

Let me calculate the exact coordinates using this setup.The coordinates of the three remaining vertices of the square are as follows:

1. Second vertex: $\left( 2\sqrt{2}, 0 \right) \approx (2.828, 0)$
2. Third vertex: $\left( \frac{2\sqrt{2}}{2}, \frac{2\sqrt{2}}{2} \right) \approx (1.414, 1.414)$
3. Fourth vertex: $\left( \frac{2\sqrt{2}}{2}, -\frac{2\sqrt{2}}{2} \right) \approx (1.414, -1.414)$

Thus, the full set of vertices of the square are:

• $(0, 0)$
• $(2.828, 0)$
• $(1.414, 1.414)$
• $(1.414, -1.414)$

Would you like a detailed breakdown of the rotation or further explanations?

Here are some related questions you can explore:

1. How can the general formula for the rotation of coordinates be applied in geometry?
2. What is the significance of the diagonal in determining the properties of a square?
3. How do we calculate the area and perimeter of this square?
4. How would the coordinates change if the square was rotated by a different angle?
5. What is the relationship between the square's side length and its diagonal?

Tip: For any square, the diagonal is always equal to $\text{side} \times \sqrt{2}$.