Find the equation of the right bisector of the line segment joining the points (3,4) and (-1,2).

Explanation

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

### Step 1: Find the Midpoint of the Line Segment
The midpoint \( M \) of the line segment joining the points \( A(3, 4) \) and \( B(-1, 2) \) 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 of points \( A \) and \( B \):
\[ M = \left( \frac{3 + (-1)}{2}, \frac{4 + 2}{2} \right) = \left( \frac{2}{2}, \frac{6}{2} \right) = (1, 3) \]

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

### Step 3: Find the Slope of the Right Bisector
The slope of the right bisector \( m_{CD} \) is the negative reciprocal of the slope of line segment \( AB \):
\[ m_{CD} = -\frac{1}{m_{AB}} = -\frac{1}{\frac{1}{2}} = -2 \]

### Step 4: Use the Point-Slope Form to Find the Equation of the Right Bisector
Now, we can use the point-slope form of the equation of a line, which is given by:
\[ y - y_1 = m(x - x_1) \]
Here, \( (x_1, y_1) \) is the midpoint \( (1, 3) \) and \( m = -2 \):
\[ y - 3 = -2(x - 1) \]
Expanding this equation:
\[ y - 3 = -2x + 2 \]
\[ y + 2x = 5 \]

### Final Equation
Thus, the equation of the right bisector of the line segment joining the points (3, 4) and (-1, 2) is:
\[ y + 2x = 5 \]

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