To determine the relationship between the two lines, we need to first calculate the slopes of each line.
### Step 1: Calculate the slope of Line 1
The formula for the slope [tex]\( m \)[/tex] of a line passing through 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]
For Line 1, the points given are [tex]\( (-2, -7) \)[/tex] and [tex]\( (-4, -7) \)[/tex]. Plugging these coordinates into the slope formula:
[tex]\[ m_1 = \frac{-7 - (-7)}{-4 - (-2)} = \frac{0}{-2} = -0.0 \][/tex]
So, the slope of Line 1 is:
[tex]\[ m_1 = -0.0\][/tex]
### Step 2: Calculate the slope of Line 2
For Line 2, the points given are [tex]\( (2, -5) \)[/tex] and [tex]\( (2, 16) \)[/tex]. Plugging these coordinates into the slope formula:
[tex]\[ m_2 = \frac{16 - (-5)}{2 - 2} \][/tex]
However, the denominator is [tex]\( 2 - 2 = 0 \)[/tex], which means the slope is undefined. In mathematical terms, this indicates that the line is vertical. We usually represent the slope of a vertical line as [tex]\( \infty \)[/tex] (infinity).
So, the slope of Line 2 is:
[tex]\[ m_2 = \infty \][/tex]
### Step 3: Determine the relationship between the lines
Now that we have the slopes:
- Slope of Line 1: [tex]\( m_1 = -0.0 \)[/tex]
- Slope of Line 2: [tex]\( m_2 = \infty \)[/tex]
To determine the relationship:
1. Parallel Lines: Two lines are parallel if their slopes are equal. Here, [tex]\( -0.0 \neq \infty \)[/tex], so the lines are not parallel.
2. Perpendicular Lines: Two lines are perpendicular if the product of their slopes is [tex]\( -1 \)[/tex]. In this case, [tex]\( -0.0 \times \infty \)[/tex] is indeterminate, meaning they are not perpendicular.
3. Neither Parallel Nor Perpendicular: Since the lines are neither parallel nor perpendicular based on the slope analysis, they fall into this category.
Thus, the final answer is:
[tex]\[ \text{Line 1 is neither parallel nor perpendicular to Line 2} \][/tex]
