To find the equation of the line that is perpendicular to the line through the points (-8, 0) and (0, 4), we first need to determine the slope of the original line.
Step 1: Calculate the slope of the original line
The slope \( m \) of a line through two points \((x_1, y_1)\) and \((x_2, y_2)\) is calculated using the formula: \[ m = \frac{y_2 - y_1}{x_2 - x_1} \]
Substituting the given points (-8, 0) and (0, 4): \[ m = \frac{4 - 0}{0 - (-8)} = \frac{4}{8} = \frac{1}{2} \]
Step 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 slope. Therefore, if the original slope is \( \frac{1}{2} \), the slope of the perpendicular line \( m' \) is: \[ m' = -\frac{1}{\left(\frac{1}{2}\right)} = -2 \]
Step 3: Write the equation of the perpendicular line
Next, we need to find the equation of the perpendicular line that shares the same y-intercept as the original line, which is at point (0, 4).
Since the y-intercept \( b \) is 4, we can write the equation of the line in slope-intercept form \( y = mx + b \): \[ y = -2x + 4 \]
Step 4: Choose the correct answer
The equation we found corresponds to option D: \[ \text{D. } y = -2x + 4 \]
Therefore, the answer is D. y = -2x + 4.