The line represented by the equation `y = x` is the perpendicular bisector of the line segment AB. If A has the coordinates (-3,3), what are the coordinates for B?

Options

A. (0,0)
B. (3,-3)
C. (0,-3)
D. (3,0)

Answer

To find the coordinates of point B given that the line \( y = x \) is the perpendicular bisector of the line segment AB, where point A has coordinates (-3, 3), we can follow these steps:

### Step 1: Find the midpoint M of segment AB
The coordinates of point A are given as \( A(-3, 3) \) and let the coordinates of point B be \( B(x_1, y_1) \). The midpoint M of segment AB can be calculated using the midpoint formula:

\[ M = \left( \frac{x_A + x_B}{2}, \frac{y_A + y_B}{2} \right) \]

Substituting the coordinates of point A and point B:

\[ M = \left( \frac{-3 + x_1}{2}, \frac{3 + y_1}{2} \right) \]

### Step 2: Set the midpoint M on the line \( y = x \)
Since M lies on the line \( y = x \), we have:

\[ \frac{3 + y_1}{2} = \frac{-3 + x_1}{2} \]

### Step 3: Simplify the equation
Multiplying both sides by 2 to eliminate the fraction:

\[ 3 + y_1 = -3 + x_1 \]

Rearranging gives us:

\[ x_1 - y_1 = 6 \quad \text{(Equation 1)} \]

### Step 4: Find the slope of line AB
The slope of line AB can be calculated using the formula for the slope between two points:

\[ m_{AB} = \frac{y_1 - y_A}{x_1 - x_A} = \frac{y_1 - 3}{x_1 + 3} \]

### Step 5: Use the property of perpendicular lines
Since the line \( y = x \) has a slope of 1, the product of the slopes of two perpendicular lines is -1:

\[ 1 \cdot \frac{y_1 - 3}{x_1 + 3} = -1 \]

This simplifies to:

\[ y_1 - 3 = - (x_1 + 3) \]

Rearranging gives us:

\[ y_1 + x_1 = 0 \quad \text{(Equation 2)} \]

### Step 6: Solve the system of equations
Now we have two equations:
1. \( x_1 - y_1 = 6 \)
2. \( y_1 + x_1 = 0 \)

From Equation 2, we can express \( y_1 \) in terms of \( x_1 \):

\[ y_1 = -x_1 \]

Substituting this into Equation 1:

\[ x_1 - (-x_1) = 6 \]

This simplifies to:

\[ 2x_1 = 6 \implies x_1 = 3 \]

### Step 7: Find \( y_1 \)
Now substituting \( x_1 = 3 \) back into Equation 2:

\[ y_1 + 3 = 0 \implies y_1 = -3 \]

### Conclusion
Thus, the coordinates of point B are:

\[ B(3, -3) \]