Point of intersection of the diagonals of square is at origin and coordinate axis are drawn along the diagonals. If the side is of length a, then one which is not the vertex of square is :

Options

A. `(a//sqrt(2),0)`
B. `(0,a//sqrt(2))`
C. `(asqrt(2),0)`
D. `(-a//sqrt(2),0)`

Answer

To solve the problem, we need to determine which of the given points is not a vertex of the square, given that the diagonals intersect at the origin and the side length of the square is \( a \).

### Step-by-Step Solution:

1. **Understanding the Square's Properties**:
- The diagonals of a square bisect each other at right angles and intersect at the center of the square.
- Given that the intersection point of the diagonals is at the origin (0, 0), this means the center of the square is at (0, 0).

2. **Finding the Length of the Diagonal**:
- The length of the diagonal \( d \) of a square can be calculated using the formula:
\[ d = a\sqrt{2} \]
- This is because the diagonal forms a right triangle with two sides of length \( a \).

3. **Calculating the Coordinates of the Vertices**:
- Since the diagonals intersect at the origin and are aligned with the coordinate axes, we can find the coordinates of the vertices.
- The distance from the origin to each vertex is half the diagonal:
\[ \text{Distance from origin to vertex} = \frac{d}{2} = \frac{a\sqrt{2}}{2} = \frac{a}{\sqrt{2}} \]
- The vertices can be determined based on their positions along the axes:
- Vertex A: \( \left(\frac{a}{\sqrt{2}}, 0\right) \)
- Vertex B: \( \left(0, \frac{a}{\sqrt{2}}\right) \)
- Vertex C: \( \left(-\frac{a}{\sqrt{2}}, 0\right) \)
- Vertex D: \( \left(0, -\frac{a}{\sqrt{2}}\right) \)

4. **Identifying the Non-Vertex Point**:
- The problem asks us to identify which of the given points is NOT a vertex of the square.
- The vertices we calculated are:
- \( \left(\frac{a}{\sqrt{2}}, 0\right) \)
- \( \left(0, \frac{a}{\sqrt{2}}\right) \)
- \( \left(-\frac{a}{\sqrt{2}}, 0\right) \)
- \( \left(0, -\frac{a}{\sqrt{2}}\right) \)

5. **Evaluating the Given Options**:
- We need to evaluate the options provided in the question:
1. \( \left(\frac{a}{\sqrt{2}}, 0\right) \) - This is Vertex A.
2. \( \left(0, \frac{a}{\sqrt{2}}\right) \) - This is Vertex B.
3. \( \left(a\sqrt{2}, 0\right) \) - This is NOT a vertex.
4. \( \left(-\frac{a}{\sqrt{2}}, 0\right) \) - This is Vertex C.

6. **Conclusion**:
- The point that is NOT a vertex of the square is \( \left(a\sqrt{2}, 0\right) \).

### Final Answer:
The point which is not a vertex of the square is \( \left(a\sqrt{2}, 0\right) \).

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