Sure! Let's carefully analyze the given problem step by step.
### Step-by-Step Solution
1. Given Slope:
The given slope of the line is [tex]\( m = \frac{17}{29} \)[/tex].
2. Finding the Slope of a Parallel Line:
- Lines that are parallel have the same slope.
- Thus, the slope of a line parallel to the given line is also [tex]\( \frac{17}{29} \)[/tex].
3. Finding the Slope of a Perpendicular Line:
- Lines that are perpendicular have slopes that are negative reciprocals of each other.
- The negative reciprocal of the given slope [tex]\( \frac{17}{29} \)[/tex] can be found as follows:
1. Take the reciprocal: [tex]\( \frac{1}{\frac{17}{29}} = \frac{29}{17} \)[/tex].
2. Make it negative: [tex]\( -\frac{29}{17} \)[/tex].
4. Identifying the Correct Option:
- We need to match these findings with the correct multiple choice option.
- Let's compare:
- Option A:
- Parallel: [tex]\( 29 \)[/tex]
- Perpendicular: [tex]\( \frac{17}{29} \)[/tex]
- This is incorrect, as the parallel slope should be [tex]\( \frac{17}{29} \)[/tex].
- Option B:
- Parallel: [tex]\( \frac{17}{29} \)[/tex]
- Perpendicular: [tex]\( -\frac{29}{17} \)[/tex]
- This matches our calculations perfectly.
- Option C:
- Parallel: [tex]\( -\frac{29}{17} \)[/tex]
- Perpendicular: [tex]\( \frac{17}{29} \)[/tex]
- This is incorrect, as the perpendicular slope should be [tex]\( -\frac{29}{17} \)[/tex] and the parallel slope should be [tex]\( \frac{17}{29} \)[/tex].
- Option D:
- Parallel: [tex]\( \frac{17}{29} \)[/tex]
- Perpendicular: [tex]\( \frac{29}{17} \)[/tex]
- This is incorrect, as the perpendicular slope should be [tex]\( -\frac{29}{17} \)[/tex].
### Conclusion
Based on the detailed analysis, option B is correct.
- Slope of a line parallel to the given line: [tex]\( \frac{17}{29} \)[/tex]
- Slope of a line perpendicular to the given line: [tex]\( -\frac{29}{17} \)[/tex]
Thus, the answer is option B.
