The equation of the perpendicular bisector of the segment joining the points whose coordinates are (1,4) and (-2,3) is

Options

A. `3x-2y+5=0`
B. `x-3y+2=0`
C. `3x+y-2=0`
D. `x=3y+11=0`

Explanation

To find the equation of the perpendicular bisector of the segment joining the points (1, 4) and (-2, 3), we will follow these steps:

### Step 1: Find the Midpoint of the Segment
The midpoint \( M \) of the segment joining points \( P(1, 4) \) and \( Q(-2, 3) \) can be calculated using the midpoint formula:
\[ M = \left( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2} \right) \]
Substituting the coordinates:
\[ M = \left( \frac{1 + (-2)}{2}, \frac{4 + 3}{2} \right) = \left( \frac{-1}{2}, \frac{7}{2} \right) \]

### Step 2: Find the Slope of Line Segment PQ
Next, we find the slope \( m_{PQ} \) of the line segment \( PQ \) using the slope formula:
\[ m_{PQ} = \frac{y_2 - y_1}{x_2 - x_1} \]
Substituting the coordinates:
\[ m_{PQ} = \frac{3 - 4}{-2 - 1} = \frac{-1}{-3} = \frac{1}{3} \]

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

### Step 4: Use Point-Slope Form to Find the Equation
Now, we will use the point-slope form of the equation of a line, which is:
\[ y - y_1 = m(x - x_1) \]
Here, \( (x_1, y_1) \) is the midpoint \( M\left(-\frac{1}{2}, \frac{7}{2}\right) \) and \( m = -3 \):
\[ y - \frac{7}{2} = -3\left(x + \frac{1}{2}\right) \]

### Step 5: Simplify the Equation
Expanding the equation:
\[ y - \frac{7}{2} = -3x - \frac{3}{2} \]
Adding \( \frac{7}{2} \) to both sides:
\[ y = -3x - \frac{3}{2} + \frac{7}{2} \]
\[ y = -3x + 2 \]

### Step 6: Rearranging to Standard Form
To write this in standard form \( Ax + By + C = 0 \):
\[ 3x + y - 2 = 0 \]
This can also be expressed as:
\[ y + 3x - 2 = 0 \]

### Final Answer
Thus, the equation of the perpendicular bisector is:
\[ y + 3x - 2 = 0 \]