The equation of the perpendicular bisector of line segment joining points A `(4,5)` and B `(-2,3)` is :

Options

A. `2x-y+7=0`
B. `3x+2y-7=0`
C. `3x-y-7=0`
D. `3x+y-7=0`

Answer

To find the equation of the perpendicular bisector of the line segment joining points A (4, 5) and B (-2, 3), we will follow these steps:

### Step 1: Find the Midpoint of AB
The midpoint \( C \) of the line segment joining points \( A(x_1, y_1) \) and \( B(x_2, y_2) \) can be calculated using the midpoint formula:
\[ C = \left( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2} \right) \]
Substituting the coordinates of points A and B:
\[ C = \left( \frac{4 + (-2)}{2}, \frac{5 + 3}{2} \right) = \left( \frac{2}{2}, \frac{8}{2} \right) = (1, 4) \]

### Step 2: Find the Slope of AB
The slope \( m_{AB} \) of the line segment AB can be calculated using the slope formula:
\[ m_{AB} = \frac{y_2 - y_1}{x_2 - x_1} \]
Substituting the coordinates of points A and B:
\[ m_{AB} = \frac{3 - 5}{-2 - 4} = \frac{-2}{-6} = \frac{1}{3} \]

### Step 3: Find the Slope of the Perpendicular Bisector
The slope of the perpendicular bisector \( m_L \) is the negative reciprocal of the slope of AB:
\[ m_L = -\frac{1}{m_{AB}} = -\frac{1}{\frac{1}{3}} = -3 \]

### Step 4: Write the Equation of the Perpendicular Bisector
Using the point-slope form of the equation of a line, which is given by:
\[ y - y_1 = m(x - x_1) \]
We can substitute \( m_L \) and the coordinates of point C:
\[ y - 4 = -3(x - 1) \]
Expanding this equation:
\[ y - 4 = -3x + 3 \]
\[ y = -3x + 7 \]

### Step 5: Rearranging to Standard Form
To express this in standard form \( Ax + By + C = 0 \):
\[ 3x + y - 7 = 0 \]

### Final Answer
The equation of the perpendicular bisector of the line segment joining points A (4, 5) and B (-2, 3) is:
\[ 3x + y - 7 = 0 \]

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