To determine which equation represents a line that is perpendicular to \( y = -2x + 4 \) and passes through the point \( (4, 2) \), we'll follow these steps:
1. Find the slope of the given line: The given line is \( y = -2x + 4 \). This is in the slope-intercept form \( y = mx + b \), where \( m \) is the slope. By comparison, the slope \( m \) of the given line is \(-2\).
2. Determine the slope of the perpendicular line: The slope of a line perpendicular to another line is the negative reciprocal of the slope of the original line. For the given slope \(-2\), the negative reciprocal is:
[tex]\[
\text{Perpendicular slope} = -\frac{1}{-2} = \frac{1}{2}
\][/tex]
3. Use the point-slope form to find the equation: The point-slope form of a line is given by:
[tex]\[
y - y_1 = m(x - x_1)
\][/tex]
where \( m \) is the slope and \( (x_1, y_1) \) is a point on the line. Here, the point \( (4, 2) \) and the slope \( \frac{1}{2} \) will be used. Substituting these values into the point-slope form yields:
[tex]\[
y - 2 = \frac{1}{2}(x - 4)
\][/tex]
4. Simplify to the slope-intercept form: We need to convert the equation to the slope-intercept form \( y = mx + b \).
[tex]\[
y - 2 = \frac{1}{2}(x - 4)
\][/tex]
Distribute the \( \frac{1}{2} \) on the right-hand side:
[tex]\[
y - 2 = \frac{1}{2}x - 2
\][/tex]
Now, add 2 to both sides to solve for \( y \):
[tex]\[
y = \frac{1}{2}x
\][/tex]
Therefore, the equation that represents a line perpendicular to \( y = -2x + 4 \) and passing through the point \( (4, 2) \) is:
[tex]\[
\boxed{y = \frac{1}{2}x}
\][/tex]
Among the given options, Choice A [tex]\( y = \frac{1}{2} x \)[/tex] is the correct answer.
