To determine whether the lines are parallel, perpendicular, or neither, we need to calculate the slopes of the lines passing through the given pairs of points and compare these slopes.
### Step-by-Step Solution:
1. Calculate the slope of Line [tex]\( a \)[/tex]:
- Line [tex]\( a \)[/tex] passes through points [tex]\((0, -1)\)[/tex] and [tex]\((5, 4)\)[/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]
- Substituting the coordinates of Line [tex]\( a \)[/tex]:
[tex]\[
m_a = \frac{4 - (-1)}{5 - 0} = \frac{4 + 1}{5} = \frac{5}{5} = 1
\][/tex]
2. Calculate the slope of Line [tex]\( b \)[/tex]:
- Line [tex]\( b \)[/tex] passes through points [tex]\((-7, 2)\)[/tex] and [tex]\((1, 10)\)[/tex].
- Using the slope formula again:
[tex]\[
m_b = \frac{10 - 2}{1 - (-7)} = \frac{10 - 2}{1 + 7} = \frac{8}{8} = 1
\][/tex]
3. Compare the slopes:
- Two lines are parallel if they have the same slope.
- Two lines are perpendicular if the product of their slopes is [tex]\(-1\)[/tex].
- If neither condition holds, the lines are neither parallel nor perpendicular.
- Since the slopes [tex]\( m_a \)[/tex] and [tex]\( m_b \)[/tex] are both equal to 1:
[tex]\[ m_a = m_b \][/tex]
This means that the lines are parallel.
### Conclusion:
The lines passing through the points [tex]\((0, -1)\)[/tex] and [tex]\((5, 4)\)[/tex], as well as [tex]\((-7, 2)\)[/tex] and [tex]\((1, 10)\)[/tex], are parallel.
