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

Options

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

Answer

To find the length of segment AB, where point A is symmetric to point B(4, -1) with respect to the bisector of the first quadrant (the line x = y), we can follow these steps:

### Step-by-Step Solution:

1. **Identify the Bisector**: The bisector of the first quadrant is the line where x = y. This line divides the first quadrant into two equal halves.

2. **Coordinates of Point B**: The coordinates of point B are given as B(4, -1).

3. **Determine the Symmetric Point A**: To find the coordinates of point A, we need to reflect point B across the line x = y. The reflection of a point (x, y) across the line x = y is given by (y, x). Therefore, reflecting point B(4, -1):
- The x-coordinate of A will be the y-coordinate of B: -1.
- The y-coordinate of A will be the x-coordinate of B: 4.
- Thus, the coordinates of point A are A(-1, 4).

4. **Calculate the Length of AB**: The length of segment AB can be calculated using the distance formula:
\[ AB = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \]
Substituting the coordinates of A and B:
\[ AB = \sqrt{((-1) - 4)^2 + (4 - (-1))^2} \]
\[ = \sqrt{(-5)^2 + (5)^2} \]
\[ = \sqrt{25 + 25} \]
\[ = \sqrt{50} \]
\[ = 5\sqrt{2} \]

5. **Final Result**: The length of AB is \(5\sqrt{2}\).