To determine whether each line is parallel, perpendicular, or neither parallel nor perpendicular to a line whose slope is [tex]\(-\frac{3}{4}\)[/tex], we need to compare the slopes. Here are the steps and definitions we use:
1. Parallel Lines: Two lines are parallel if they have the same slope.
2. Perpendicular Lines: Two lines are perpendicular if the product of their slopes is [tex]\(-1\)[/tex].
3. Neither: If the lines are neither parallel nor perpendicular, they fall into this category.
Given slopes:
- Line [tex]\( m \)[/tex] with slope [tex]\(\frac{3}{4}\)[/tex]
- Line [tex]\( n \)[/tex] with slope [tex]\(\frac{4}{3}\)[/tex]
- Line [tex]\( p \)[/tex] with slope [tex]\(-\frac{4}{3}\)[/tex]
- Line [tex]\( q \)[/tex] with slope [tex]\(-\frac{3}{4}\)[/tex]
We compare each slope with the given slope [tex]\(-\frac{3}{4}\)[/tex]:
### Line [tex]\( m \)[/tex] with slope [tex]\(\frac{3}{4}\)[/tex]
- Parallel: [tex]\(\frac{3}{4} \ne -\frac{3}{4}\)[/tex]
- Perpendicular: [tex]\(\frac{3}{4} \times \left(-\frac{3}{4}\right) = -\frac{9}{16} \ne -1\)[/tex]
- Therefore, Line [tex]\( m \)[/tex] is Neither parallel nor perpendicular.
### Line [tex]\( n \)[/tex] with slope [tex]\(\frac{4}{3}\)[/tex]
- Parallel: [tex]\(\frac{4}{3} \ne -\frac{3}{4}\)[/tex]
- Perpendicular: [tex]\(\frac{4}{3} \times \left(-\frac{3}{4}\right) = -1\)[/tex]
- Therefore, Line [tex]\( n \)[/tex] is Perpendicular.
### Line [tex]\( p \)[/tex] with slope [tex]\(-\frac{4}{3}\)[/tex]
- Parallel: [tex]\(-\frac{4}{3} \ne -\frac{3}{4}\)[/tex]
- Perpendicular: [tex]\(-\frac{4}{3} \times \left(-\frac{3}{4}\right) = \frac{16}{9} \ne -1\)[/tex]
- Therefore, Line [tex]\( p \)[/tex] is Neither parallel nor perpendicular.
### Line [tex]\( q \)[/tex] with slope [tex]\(-\frac{3}{4}\)[/tex]
- Parallel: [tex]\(-\frac{3}{4} = -\frac{3}{4}\)[/tex]
- Therefore, Line [tex]\( q \)[/tex] is Parallel.
So, the completed table should be:
- Line [tex]\( m \)[/tex]: Neither
- Line [tex]\( n \)[/tex]: Perpendicular
- Line [tex]\( p \)[/tex]: Neither
- Line [tex]\( q \)[/tex]: Parallel
