Which of the following represents an equation of the line that is the perpendicular bisector of the segment whose endpoints are `(-2, 4) and (8, 4)`?

Options

A. `x=3`
B. `y=3`
C. `x=5`
D. `y=5`

Answer

To find the equation of the perpendicular bisector of the line segment whose endpoints are \((-2, 4)\) and \((8, 4)\), we can follow these steps:

### Step 1: Identify the Midpoint of the Segment
The midpoint \(M\) of a line segment with endpoints \((x_1, y_1)\) and \((x_2, y_2)\) can be calculated using the formula:
\[ M = \left(\frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2}\right) \]
For our endpoints \((-2, 4)\) and \((8, 4)\):
- \(x_1 = -2\), \(y_1 = 4\)
- \(x_2 = 8\), \(y_2 = 4\)

Calculating the midpoint:
\[ M = \left(\frac{-2 + 8}{2}, \frac{4 + 4}{2}\right) = \left(\frac{6}{2}, \frac{8}{2}\right) = (3, 4) \]

### Step 2: Determine the Slope of the Original Line Segment
The slope \(m\) of a horizontal line (which is the case here since both endpoints have the same \(y\)-coordinate) is:
\[ m = \frac{y_2 - y_1}{x_2 - x_1} = \frac{4 - 4}{8 - (-2)} = \frac{0}{10} = 0 \]

### Step 3: Find the Slope of the Perpendicular Bisector
The slope of a line that is perpendicular to another line is the negative reciprocal of the original line's slope. Since the slope of our line segment is \(0\), the slope of the perpendicular bisector will be undefined, indicating that it is a vertical line.

### Step 4: Write the Equation of the Perpendicular Bisector
Since the perpendicular bisector passes through the midpoint \((3, 4)\) and is vertical, its equation is:
\[ x = 3 \]

### Conclusion
The equation of the perpendicular bisector of the segment with endpoints \((-2, 4)\) and \((8, 4)\) is:
\[ \boxed{x = 3} \]