To solve this problem, we need to determine the equation of a line that is perpendicular to the equation \( y = -2x + 4 \) and passes through the point \( (4, 2) \).
### Step-by-Step Solution:
1. Find the slope of the given line:
The given equation of the line is \( y = -2x + 4 \). The slope of this line (denoted as \( m \)) is -2.
2. Determine the slope of the perpendicular line:
The slope of the line perpendicular to another line is the negative reciprocal of the slope of the original line.
- The negative reciprocal of \(-2\) is \(\frac{1}{2}\).
So, the slope of the perpendicular line is \(\frac{1}{2}\).
3. Identify possible equations:
Given the options:
- \( y = \frac{1}{2} x \)
- \( y = \frac{1}{2} x + 4\)
- \( y = -\frac{1}{2} x + 2 \)
- \( y = -2 x \)
We see that options A and B have the correct slope \(\frac{1}{2}\) for the perpendicular line. However, we still need to find which specific line passes through the point \( (4, 2) \).
4. Check which line passes through \((4, 2)\):
- Option A: \( y = \frac{1}{2} x \)
- Substitute \( x = 4 \): \( y = \frac{1}{2} (4) = 2 \).
- This results in the point \((4, 2)\). So, this line passes through the point.
- Option B: \( y = \frac{1}{2} x + 4\)
- Substitute \( x = 4 \): \( y = \frac{1}{2} (4) + 4 = 2 + 4 = 6 \).
- This does not result in the point \((4, 2)\).
- Option C: \( y = -\frac{1}{2} x + 2 \)
- The slope of this line is \(-\frac{1}{2}\), which is incorrect for being perpendicular.
- Option D: \( y = -2 x\)
- The slope of this line is \(-2\), which is incorrect for being perpendicular.
By examining each option, we conclude that:
- Option A: \( y = \frac{1}{2} x \) is the equation that is perpendicular to \( y = -2x + 4 \) and passes through the point \( (4, 2) \).
Hence, the correct equation is:
A. [tex]\( y = \frac{1}{2} x \)[/tex].
