If the point `A` is symmetric to the point `B(4,-1)` with respect to the bisector of the first quadrant then `AB` is

Options

A. `3sqrt(2)`
B. `5sqrt(2)`
C. `7sqrt(2)`
D. `9sqrt(2)`

Explanation

To solve the problem, we need to find the distance \( AB \) where point \( A \) is symmetric to point \( B(4, -1) \) with respect to the bisector of the first quadrant, which is the line \( y = x \).

### Step-by-Step Solution:

1. **Identify the coordinates of point B**:
Given point \( B \) is \( (4, -1) \).

2. **Determine the symmetric point A**:
The line \( y = x \) bisects the first quadrant. To find the symmetric point \( A \) of point \( B \) with respect to this line, we swap the x-coordinate and y-coordinate of point \( B \). Thus, the coordinates of point \( A \) will be:
\[ A = (-1, 4) \]

3. **Use the distance formula to find AB**:
The distance \( AB \) between points \( A(x_1, y_1) \) and \( B(x_2, y_2) \) can be calculated using the distance formula:
\[ AB = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \]
Here, \( A(-1, 4) \) and \( B(4, -1) \). So, substituting the coordinates:
\[ AB = \sqrt{(4 - (-1))^2 + (-1 - 4)^2} \]
Simplifying the expression:
\[ AB = \sqrt{(4 + 1)^2 + (-1 - 4)^2} \]
\[ AB = \sqrt{(5)^2 + (-5)^2} \]
\[ AB = \sqrt{25 + 25} \]
\[ AB = \sqrt{50} \]
\[ AB = 5\sqrt{2} \]

4. **Final Answer**:
The distance \( AB \) is \( 5\sqrt{2} \).