To determine the relationship between the two given lines, we'll need to calculate their slopes and then use these slopes to make a conclusion about their relationship, such as whether they are parallel, perpendicular, or neither.
### Step-by-Step Solution
1. Identify the coordinates for each line:
- Line 1 passes through the points \((9, -4)\) and \((9, -1)\).
- Line 2 passes through the points \((-6, 4)\) and \((-2, 4)\).
2. Calculate the slope of Line 1:
The slope formula is \(\text{slope} = \frac{y_2 - y_1}{x_2 - x_1}\).
For Line 1:
- \((x_1, y_1) = (9, -4)\)
- \((x_2, y_2) = (9, -1)\)
Plugging in these values:
[tex]\[
\text{slope} = \frac{-1 - (-4)}{9 - 9} = \frac{3}{0}
\][/tex]
Since division by zero is undefined, the slope of Line 1 is undefined. This corresponds to Line 1 being a vertical line.
3. Calculate the slope of Line 2:
Using the same formula for slope:
- \((x_1, y_1) = (-6, 4)\)
- \((x_2, y_2) = (-2, 4)\)
Plugging in these values:
[tex]\[
\text{slope} = \frac{4 - 4}{-2 - (-6)} = \frac{0}{4} = 0
\][/tex]
The slope of Line 2 is \(0\). This corresponds to Line 2 being a horizontal line.
4. Determine the relationship between the lines:
- Line 1 has an undefined slope (vertical line).
- Line 2 has a slope of \(0\) (horizontal line).
Vertical lines and horizontal lines are always perpendicular to each other because their slopes are negative reciprocals (or one is zero and the other is undefined).
Therefore, the relationship between Line 1 and Line 2 is that:
[tex]\[
\text{Line 1 is perpendicular to Line 2}.
\][/tex]
