Find the equation of perpendicular bisector of the line made by joining the points (1, 1) and (3, 5).

Options

A. `x+2y+8=0`
B. `x-2y+8=0`
C. `x-2y-8=0`
D. `x+2y-8=0`

Answer

To find the equation of the perpendicular bisector of the line segment joining the points (1, 1) and (3, 5), we will follow these steps:

### Step 1: Find the Midpoint of the Line Segment
The midpoint \( M \) of the line segment joining the points \( A(1, 1) \) and \( B(3, 5) \) 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 \( B \):
\[ M = \left( \frac{1 + 3}{2}, \frac{1 + 5}{2} \right) = \left( \frac{4}{2}, \frac{6}{2} \right) = (2, 3) \]

### Step 2: Find the Slope of the Line Segment
Next, we need to find the slope \( m \) of the line segment \( AB \) using the slope formula:
\[ m = \frac{y_2 - y_1}{x_2 - x_1} \]
Substituting the coordinates of points \( A \) and \( B \):
\[ m = \frac{5 - 1}{3 - 1} = \frac{4}{2} = 2 \]

### Step 3: Find the Slope of the Perpendicular Bisector
The slope of the perpendicular bisector is the negative reciprocal of the slope of the line segment. Thus,
\[ m_{perpendicular} = -\frac{1}{m} = -\frac{1}{2} \]

### Step 4: Write the Equation of the Perpendicular Bisector
Now we can use the point-slope form of the equation of a line, which is given by:
\[ y - y_1 = m(x - x_1) \]
Using the midpoint \( (2, 3) \) and the slope \( -\frac{1}{2} \):
\[ y - 3 = -\frac{1}{2}(x - 2) \]
Distributing the slope:
\[ y - 3 = -\frac{1}{2}x + 1 \]
Adding 3 to both sides:
\[ y = -\frac{1}{2}x + 4 \]

### Step 5: Rearranging to Standard Form
To express the equation in standard form \( Ax + By + C = 0 \):
\[ \frac{1}{2}x + y - 4 = 0 \]
Multiplying through by 2 to eliminate the fraction:
\[ x + 2y - 8 = 0 \]

Thus, the equation of the perpendicular bisector is:
\[ x + 2y - 8 = 0 \]