Line \( P \) passes through \( (-2,5) \) and \( (4,-7) \). Write an equation for the line that passes through \( (2,9) \) and is perpendicular to line \( P \). Leave in slope-intercept form.

To find the equation of the line that is perpendicular to line \( P \) and passes through the point \( (2,9) \), we first need to determine the slope of line \( P \). 1. **Calculate the slope of line \( P \)**: The slope \( m \) of a line passing through two points \( (x_1, y_1) \) and \( (x_2, y_2) \) is given by the formula: \[ m = \frac{y_2 - y_1}{x_2 - x_1} \] For the points \( (-2, 5) \) and \( (4, -7) \): \[ m_P = \frac{-7 - 5}{4 - (-2)} = \frac{-12}{6} = -2 \] 2. **Find 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. Therefore, the slope \( m_{\perp} \) of the line we want is: \[ m_{\perp} = -\frac{1}{m_P} = -\frac{1}{-2} = \frac{1}{2} \] 3. **Use the point-slope form to find the equation**: We can use the point-slope form of the equation of a line, which is: \[ y - y_1 = m(x - x_1) \] where \( (x_1, y_1) \) is the point through which the line passes, and \( m \) is the slope. Here, we have the point \( (2, 9) \) and the slope \( \frac{1}{2} \): \[ y - 9 = \frac{1}{2}(x - 2) \] 4. **Simplify to slope-intercept form**: Distributing the slope on the right side: \[ y - 9 = \frac{1}{2}x - 1 \] Now, add 9 to both sides: \[ y = \frac{1}{2}x - 1 + 9 \] \[ y = \frac{1}{2}x + 8 \] Thus, the equation of the line that passes through \( (2, 9) \) and is perpendicular to line \( P \) is: \[ \boxed{y = \frac{1}{2}x + 8} \]