Question 16 What is the slope of the line perpendicular to the line that passes through the points \( (5,-2) \) and \( (3,6) \) ? Submit Question

To find the slope of the line perpendicular to the line that passes through the points \( (5,-2) \) and \( (3,6) \), we will follow these steps: 1. **Calculate the slope of the line through the given points.** The formula for the slope \( m \) between two points \( (x_1, y_1) \) and \( (x_2, y_2) \) is given by: \[ m = \frac{y_2 - y_1}{x_2 - x_1} \] 2. **Determine the slope of the perpendicular line.** The slope of a line that is perpendicular to another line is the negative reciprocal of the original line's slope. Let's start by calculating the slope of the line through the points \( (5,-2) \) and \( (3,6) \). Let \( (x_1, y_1) = (5, -2) \) and \( (x_2, y_2) = (3, 6) \). Now, we can calculate the slope \( m \). \[ m = \frac{6 - (-2)}{3 - 5} = \frac{6 + 2}{3 - 5} = \frac{8}{-2} = -4 \] Now that we have the slope of the original line, we can find the slope of the line that is perpendicular to it. The slope of the perpendicular line \( m_{\perp} \) is given by: \[ m_{\perp} = -\frac{1}{m} \] Substituting the value of \( m \): \[ m_{\perp} = -\frac{1}{-4} = \frac{1}{4} \] Thus, the slope of the line perpendicular to the line that passes through the points \( (5,-2) \) and \( (3,6) \) is \( \frac{1}{4} \).