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 determine the slope of line \( P \), we calculate it using the formula: \[ m = \frac{y_2 - y_1}{x_2 - x_1} \] Substituting the given points \( (-2, 5) \) and \( (4, -7) \): \[ m_P = \frac{-7 - 5}{4 - (-2)} = \frac{-12}{6} = -2 \] The slope of line \( P \) is \( -2 \). The slope of a line perpendicular to line \( P \) is the negative reciprocal of \( -2 \), which is \( \frac{1}{2} \). Now, we use the point-slope form of the equation of a line, which is: \[ y - y_1 = m(x - x_1) \] Using the point \( (2, 9) \) and slope \( \frac{1}{2} \): \[ y - 9 = \frac{1}{2}(x - 2) \] Expanding and simplifying: \[ y - 9 = \frac{1}{2}x - 1 \] \[ y = \frac{1}{2}x + 8 \] Thus, the equation of the line that passes through \( (2, 9) \) and is perpendicular to line \( P \) is: \[ y = \frac{1}{2}x + 8 \] First additional information: The concept of perpendicular lines is a foundational element in geometry, intertwining slopes, angles, and even graphing! Remember, two lines are perpendicular if the product of their slopes equals -1. So, if you ever find yourself with one slope, simply take the negative reciprocal to discover the slope of the perpendicular line. Second additional information: This equation is already in slope-intercept form, which is fantastic for quickly determining the line's slope and y-intercept! You can easily graph it by starting at the y-intercept (0,8) and rising 1 unit up for every 2 units you move to the right. It’s a fun way to see how one line can set the stage for another!