Find the slope of a line perpendicular to the line which passes through each pair of the following points: (1, -11) and (5, 2)

Answer

To find the slope of a line that is perpendicular to the line passing through the points (1, -11) and (5, 2), we will follow these steps:

### Step 1: Identify the coordinates
We have two points:
- Point 1 (x1, y1) = (1, -11)
- Point 2 (x2, y2) = (5, 2)

### Step 2: Use the slope formula
The formula for the slope (m) of a line passing through two points is given by:
\[ m = \frac{y_2 - y_1}{x_2 - x_1} \]

### Step 3: Substitute the values into the formula
Substituting the coordinates into the slope formula:
\[ m = \frac{2 - (-11)}{5 - 1} \]

### Step 4: Simplify the expression
Calculating the numerator and denominator separately:
- Numerator: \(2 - (-11) = 2 + 11 = 13\)
- Denominator: \(5 - 1 = 4\)

Thus, we have:
\[ m = \frac{13}{4} \]

### Step 5: Find the slope of the perpendicular line
For two lines to be perpendicular, the product of their slopes must equal -1. If \(m_1\) is the slope of the original line, then the slope \(m_2\) of the line perpendicular to it can be found using:
\[ m_1 \cdot m_2 = -1 \]
Substituting \(m_1 = \frac{13}{4}\):
\[ \frac{13}{4} \cdot m_2 = -1 \]

### Step 6: Solve for \(m_2\)
To find \(m_2\):
\[ m_2 = -\frac{4}{13} \]

### Final Answer
The slope of the line that is perpendicular to the line passing through the points (1, -11) and (5, 2) is:
\[ \boxed{-\frac{4}{13}} \]