The coordinates of the centre of the circle inscribed in the square formed by the lines x=2, x=6, y=5 and y=9 is:

Options

A. A. (4, 7)
B. B. (7, 4)
C. C. (3, 6)
D. D. (9, 5)

Answer

To find the coordinates of the center of the circle inscribed in the square formed by the lines \( x = 2 \), \( x = 6 \), \( y = 5 \), and \( y = 9 \), we can follow these steps:

### Step 1: Identify the vertices of the square
The lines given form a rectangle (which is also a square in this case). The vertices can be determined by finding the intersection points of the lines:
- The intersection of \( x = 2 \) and \( y = 5 \) gives the point \( (2, 5) \).
- The intersection of \( x = 2 \) and \( y = 9 \) gives the point \( (2, 9) \).
- The intersection of \( x = 6 \) and \( y = 5 \) gives the point \( (6, 5) \).
- The intersection of \( x = 6 \) and \( y = 9 \) gives the point \( (6, 9) \).

So, the vertices of the square are:
1. \( (2, 5) \)
2. \( (2, 9) \)
3. \( (6, 5) \)
4. \( (6, 9) \)

### Step 2: Find the center of the square
The center of the square can be found by calculating the midpoint of the diagonal formed by any two opposite vertices. We can use the points \( (2, 5) \) and \( (6, 9) \).

Using the midpoint formula:
\[ \text{Midpoint} = \left( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2} \right) \]
where \( (x_1, y_1) = (2, 5) \) and \( (x_2, y_2) = (6, 9) \).

Calculating the x-coordinate:
\[ x = \frac{2 + 6}{2} = \frac{8}{2} = 4 \]

Calculating the y-coordinate:
\[ y = \frac{5 + 9}{2} = \frac{14}{2} = 7 \]

Thus, the coordinates of the center of the square are \( (4, 7) \).

### Step 3: Conclusion
The coordinates of the center of the circle inscribed in the square are \( (4, 7) \).

### Final Answer:
The coordinates of the center of the circle inscribed in the square are \( (4, 7) \).

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