Find the equation of the perpendicular bisector of the st.line segment whose end points are `(0,5)` and `(-4,1)`.

Answer

To find the equation of the perpendicular bisector of the line segment whose endpoints are (0, 5) and (-4, 1), we will follow these steps:

### Step 1: Find the Midpoint of the Line Segment
The midpoint \( M \) of a line segment with endpoints \( (x_1, y_1) \) and \( (x_2, y_2) \) is given by the formula:
\[ M = \left( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2} \right) \]
Here, \( (x_1, y_1) = (0, 5) \) and \( (x_2, y_2) = (-4, 1) \).

Calculating the midpoint:
\[ M = \left( \frac{0 + (-4)}{2}, \frac{5 + 1}{2} \right) = \left( \frac{-4}{2}, \frac{6}{2} \right) = (-2, 3) \]

### Step 2: Find the Slope of the Line Segment
The slope \( m \) of a line passing through two points \( (x_1, y_1) \) and \( (x_2, y_2) \) is given by:
\[ m = \frac{y_2 - y_1}{x_2 - x_1} \]
Using our points:
\[ m = \frac{1 - 5}{-4 - 0} = \frac{-4}{-4} = 1 \]

### 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 the line segment:
\[ m_{\perpendicular} = -\frac{1}{m} = -\frac{1}{1} = -1 \]

### Step 4: Use the Point-Slope Form to Find the Equation
The point-slope form of a line is given by:
\[ y - y_1 = m(x - x_1) \]
Using the midpoint \( (-2, 3) \) and the slope \( -1 \):
\[ y - 3 = -1(x + 2) \]

### Step 5: Simplify the Equation
Expanding and simplifying:
\[ y - 3 = -x - 2 \]
\[ y = -x + 1 \]

### Step 6: Rearranging to Standard Form
To express the equation in standard form:
\[ x + y - 1 = 0 \]

Thus, the equation of the perpendicular bisector is:
\[ \boxed{x + y - 1 = 0} \]

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