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Is each line parallel, perpendicular, or neither parallel nor perpendicular to a line whose slope is [tex]-\frac{3}{4}[/tex]?

Drag each choice into the boxes to correctly complete the table.

| Line | Slope | Relation to [tex]-\frac{3}{4}[/tex] |
|------|-----------------|-------------------------------------|
| [tex]$m$[/tex] | [tex]$\frac{3}{4}$[/tex] | |
| [tex]$n$[/tex] | [tex]$\frac{4}{3}$[/tex] | |
| [tex]$p$[/tex] | [tex]$-\frac{4}{3}$[/tex] | |
| [tex]$q$[/tex] | [tex]$-\frac{3}{4}$[/tex] | |

Choices:
- Parallel
- Perpendicular
- Neither

Sagot :

GinnyAnswer
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
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