Each side of a square of length 6 and its centre is at the point (4,5). If one of its diagonals is parallel to the line y=x then the co ordinates of the vertics of the square are……………….

Answer

To find the coordinates of the vertices of the square, we can follow these steps:

### Step 1: Understand the properties of the square
Given that the square has a side length of 6 and its center is at the point (4,5), we know that the distance from the center to each vertex is half the diagonal length. The diagonal of a square can be calculated using the formula:
\[ \text{Diagonal} = \text{side} \times \sqrt{2} \]
Thus, the diagonal length is:
\[ \text{Diagonal} = 6 \times \sqrt{2} = 6\sqrt{2} \]
The distance from the center to each vertex (half the diagonal) is:
\[ \frac{6\sqrt{2}}{2} = 3\sqrt{2} \]

### Step 2: Determine the orientation of the square
Since one diagonal is parallel to the line \(y = x\), the other diagonal will be parallel to the line \(y = -x\). This means that the square is rotated 45 degrees from the axes.

### Step 3: Calculate the coordinates of the vertices
To find the vertices, we can use the center (4,5) and move \(3\sqrt{2}\) units in the directions of the diagonals.

1. **Vertex A**: Move along the line \(y = x\):
\[ A = \left(4 + 3, 5 + 3\right) = (7, 8) \]
2. **Vertex B**: Move along the line \(y = -x\):
\[ B = \left(4 + 3, 5 - 3\right) = (7, 2) \]
3. **Vertex C**: Move in the opposite direction along the line \(y = x\):
\[ C = \left(4 - 3, 5 + 3\right) = (1, 8) \]
4. **Vertex D**: Move in the opposite direction along the line \(y = -x\):
\[ D = \left(4 - 3, 5 - 3\right) = (1, 2) \]

### Step 4: List the vertices
The coordinates of the vertices of the square are:
- A (7, 8)
- B (7, 2)
- C (1, 8)
- D (1, 2)

### Final Answer
The coordinates of the vertices of the square are:
\[ (7, 8), (7, 2), (1, 8), (1, 2) \]