Let's determine the relationship between the lines that pass through the given points.
### Step 1: Find the slope of each line
The slope of a line passing through two points [tex]\((x_1, y_1)\)[/tex] and [tex]\((x_2, y_2)\)[/tex] is given by the formula:
[tex]\[ \text{slope} = \frac{y_2 - y_1}{x_2 - x_1} \][/tex]
#### Line [tex]\(a\)[/tex]
Points: [tex]\((-5, 2)\)[/tex] and [tex]\((1, 6)\)[/tex]
Calculate the slope:
[tex]\[ \text{slope}_a = \frac{6 - 2}{1 - (-5)} = \frac{4}{1 + 5} = \frac{4}{6} = \frac{2}{3} \][/tex]
#### Line [tex]\(b\)[/tex]
Points: [tex]\((-4, -2)\)[/tex] and [tex]\((2, 2)\)[/tex]
Calculate the slope:
[tex]\[ \text{slope}_b = \frac{2 - (-2)}{2 - (-4)} = \frac{2 + 2}{2 + 4} = \frac{4}{6} = \frac{2}{3} \][/tex]
### Step 2: Compare the slopes to determine the relationship
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].
3. Neither: If neither condition is met, then the lines are neither parallel nor perpendicular.
#### Check for parallel lines:
Since [tex]\(\text{slope}_a = \text{slope}_b = \frac{2}{3}\)[/tex], the slopes are equal.
Hence, the two lines are parallel.
### Final Answer:
[tex]\[ \boxed{\text{Parallel}} \][/tex]
