The equation of the right bisector of the line segment joining the points (7,4) and(-1,-2) is

Options

A. `4x+3y-10=0`
B. `3x-4y+7=0`
C. `4x+3y-15=0`
D. none

Answer

To find the equation of the right bisector of the line segment joining the points \( P(7, 4) \) and \( Q(-1, -2) \), we will follow these steps:

### Step 1: Find the Midpoint of the Line Segment
The midpoint \( R \) of the line segment joining points \( P(x_1, y_1) \) and \( Q(x_2, y_2) \) can be calculated using the formula:
\[ R = \left( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2} \right) \]
Substituting the coordinates of points \( P(7, 4) \) and \( Q(-1, -2) \):
\[ R = \left( \frac{7 + (-1)}{2}, \frac{4 + (-2)}{2} \right) = \left( \frac{6}{2}, \frac{2}{2} \right) = (3, 1) \]

### Step 2: Find the Slope of the Line Segment
The slope \( m \) of the line segment joining points \( P \) and \( Q \) is given by:
\[ m = \frac{y_2 - y_1}{x_2 - x_1} \]
Substituting the coordinates:
\[ m = \frac{-2 - 4}{-1 - 7} = \frac{-6}{-8} = \frac{3}{4} \]

### Step 3: Find the Slope of the Right Bisector
The slope of the right bisector is the negative reciprocal of the slope of the line segment. Therefore:
\[ m_{bisector} = -\frac{1}{m} = -\frac{1}{\frac{3}{4}} = -\frac{4}{3} \]

### Step 4: Write the Equation of the Right Bisector
Using the point-slope form of the equation of a line, which is:
\[ y - y_1 = m(x - x_1) \]
where \( (x_1, y_1) \) is the midpoint \( R(3, 1) \) and \( m = -\frac{4}{3} \):
\[ y - 1 = -\frac{4}{3}(x - 3) \]

### Step 5: Simplify the Equation
Expanding the equation:
\[ y - 1 = -\frac{4}{3}x + 4 \]
Adding 1 to both sides:
\[ y = -\frac{4}{3}x + 5 \]

### Step 6: Convert to Standard Form
To convert the equation to standard form \( Ax + By + C = 0 \):
\[ \frac{4}{3}x + y - 5 = 0 \]
Multiplying through by 3 to eliminate the fraction:
\[ 4x + 3y - 15 = 0 \]

Thus, the equation of the right bisector is:
\[ 4x + 3y - 15 = 0 \]

### Final Answer
The equation of the right bisector of the line segment joining the points \( (7, 4) \) and \( (-1, -2) \) is:
\[ 4x + 3y - 15 = 0 \]

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