Answer :
To determine the relationship between the two lines passing through the given pairs of points, we need to analyze their slopes.
### Finding the Slope of Line [tex]\(a\)[/tex]
Line [tex]\(a\)[/tex] passes through the points [tex]\((0, 4)\)[/tex] and [tex]\((6, 0)\)[/tex]. The formula for the slope ([tex]\(m\)[/tex]) between two points [tex]\((x_1, y_1)\)[/tex] and [tex]\((x_2, y_2)\)[/tex] is:
[tex]\[ m = \frac{y_2 - y_1}{x_2 - x_1} \][/tex]
So, for line [tex]\(a\)[/tex]:
[tex]\[ m_a = \frac{0 - 4}{6 - 0} = \frac{-4}{6} = -\frac{2}{3} \][/tex]
### Finding the Slope of Line [tex]\(b\)[/tex]
Line [tex]\(b\)[/tex] passes through the points [tex]\((4, -1)\)[/tex] and [tex]\((6, 2)\)[/tex]. Using the same slope formula:
[tex]\[ m_b = \frac{2 - (-1)}{6 - 4} = \frac{2 + 1}{6 - 4} = \frac{3}{2} \][/tex]
### Comparing the Slopes
1. Parallel Lines: Two lines are parallel if their slopes are equal.
2. Perpendicular Lines: Two lines are perpendicular if the product of their slopes is [tex]\(-1\)[/tex].
Let's compare the slopes:
- Slope of line [tex]\(a\)[/tex]: [tex]\( -\frac{2}{3}\)[/tex]
- Slope of line [tex]\(b\)[/tex]: [tex]\( \frac{3}{2}\)[/tex]
Calculate the product of the slopes:
[tex]\[ m_a \times m_b = \left(-\frac{2}{3}\right) \times \left(\frac{3}{2}\right) = -1 \][/tex]
Since the product of the slopes is [tex]\(-1\)[/tex], the lines are perpendicular.
### Conclusion
Based on the slope calculations and their relationship, the lines passing through the given pairs of points are perpendicular.
So, the correct answer is:
Perpendicular
### Finding the Slope of Line [tex]\(a\)[/tex]
Line [tex]\(a\)[/tex] passes through the points [tex]\((0, 4)\)[/tex] and [tex]\((6, 0)\)[/tex]. The formula for the slope ([tex]\(m\)[/tex]) between two points [tex]\((x_1, y_1)\)[/tex] and [tex]\((x_2, y_2)\)[/tex] is:
[tex]\[ m = \frac{y_2 - y_1}{x_2 - x_1} \][/tex]
So, for line [tex]\(a\)[/tex]:
[tex]\[ m_a = \frac{0 - 4}{6 - 0} = \frac{-4}{6} = -\frac{2}{3} \][/tex]
### Finding the Slope of Line [tex]\(b\)[/tex]
Line [tex]\(b\)[/tex] passes through the points [tex]\((4, -1)\)[/tex] and [tex]\((6, 2)\)[/tex]. Using the same slope formula:
[tex]\[ m_b = \frac{2 - (-1)}{6 - 4} = \frac{2 + 1}{6 - 4} = \frac{3}{2} \][/tex]
### Comparing the Slopes
1. Parallel Lines: Two lines are parallel if their slopes are equal.
2. Perpendicular Lines: Two lines are perpendicular if the product of their slopes is [tex]\(-1\)[/tex].
Let's compare the slopes:
- Slope of line [tex]\(a\)[/tex]: [tex]\( -\frac{2}{3}\)[/tex]
- Slope of line [tex]\(b\)[/tex]: [tex]\( \frac{3}{2}\)[/tex]
Calculate the product of the slopes:
[tex]\[ m_a \times m_b = \left(-\frac{2}{3}\right) \times \left(\frac{3}{2}\right) = -1 \][/tex]
Since the product of the slopes is [tex]\(-1\)[/tex], the lines are perpendicular.
### Conclusion
Based on the slope calculations and their relationship, the lines passing through the given pairs of points are perpendicular.
So, the correct answer is:
Perpendicular
