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 solve the problem of finding 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 y = x), we can follow these steps:

### Step 1: Understand the Symmetry
The bisector of the first quadrant is the line y = x. When a point is reflected across this line, the x and y coordinates of the point are swapped.

### Step 2: Identify Point B
We have point B given as B(4, -1).

### Step 3: Reflect Point B to Find Point A
To find point A, we need to swap the coordinates of point B:
- The x-coordinate of B is 4, and the y-coordinate is -1.
- After reflection, the coordinates of point A will be A(-1, 4).

### Step 4: Calculate the Length of AB
To find the distance between points A and B, we can use the distance formula:
\[ AB = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \]
Substituting the coordinates of points A and B:
- A(-1, 4) and B(4, -1)
\[ AB = \sqrt{(4 - (-1))^2 + (-1 - 4)^2} \]
\[ AB = \sqrt{(4 + 1)^2 + (-5)^2} \]
\[ AB = \sqrt{(5)^2 + (-5)^2} \]
\[ AB = \sqrt{25 + 25} \]
\[ AB = \sqrt{50} \]
\[ AB = 5\sqrt{2} \]

### Final Answer
The length of AB is \( 5\sqrt{2} \).

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