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Answer :
To determine whether the lines [tex]\( L_1 \)[/tex] and [tex]\( L_2 \)[/tex] are parallel, perpendicular, or neither, we need to compare their slopes.
### Step 1: Find the Slope of Line [tex]\( L_1 \)[/tex]
The formula to find 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 [tex]\( L_1 \)[/tex], the points are [tex]\((1, 2)\)[/tex] and [tex]\((3, 1)\)[/tex].
Plug these values into the slope formula:
[tex]\[ m_1 = \frac{1 - 2}{3 - 1} = \frac{-1}{2} = -0.5 \][/tex]
### Step 2: Find the Slope of Line [tex]\( L_2 \)[/tex]
For line [tex]\( L_2 \)[/tex], the points are [tex]\((0, -1)\)[/tex] and [tex]\((2, 0)\)[/tex].
Again, use the slope formula:
[tex]\[ m_2 = \frac{0 - (-1)}{2 - 0} = \frac{1}{2} = 0.5 \][/tex]
### Step 3: Compare the Slopes
- Parallel Lines: Two lines are parallel if their slopes are equal. Here, [tex]\( m_1 = -0.5 \)[/tex] and [tex]\( m_2 = 0.5 \)[/tex]; they are not equal, so the lines are not parallel.
- Perpendicular Lines: Two lines are perpendicular if the product of their slopes is [tex]\(-1\)[/tex]. Calculate the product:
[tex]\[ m_1 \times m_2 = (-0.5) \times (0.5) = -0.25 \][/tex]
Since [tex]\(-0.25\)[/tex] is not equal to [tex]\(-1\)[/tex], the lines are not perpendicular.
### Conclusion
The lines [tex]\( L_1 \)[/tex] and [tex]\( L_2 \)[/tex] are neither parallel nor perpendicular because the conditions for either relationship are not satisfied based on their slopes.
### Step 1: Find the Slope of Line [tex]\( L_1 \)[/tex]
The formula to find 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 [tex]\( L_1 \)[/tex], the points are [tex]\((1, 2)\)[/tex] and [tex]\((3, 1)\)[/tex].
Plug these values into the slope formula:
[tex]\[ m_1 = \frac{1 - 2}{3 - 1} = \frac{-1}{2} = -0.5 \][/tex]
### Step 2: Find the Slope of Line [tex]\( L_2 \)[/tex]
For line [tex]\( L_2 \)[/tex], the points are [tex]\((0, -1)\)[/tex] and [tex]\((2, 0)\)[/tex].
Again, use the slope formula:
[tex]\[ m_2 = \frac{0 - (-1)}{2 - 0} = \frac{1}{2} = 0.5 \][/tex]
### Step 3: Compare the Slopes
- Parallel Lines: Two lines are parallel if their slopes are equal. Here, [tex]\( m_1 = -0.5 \)[/tex] and [tex]\( m_2 = 0.5 \)[/tex]; they are not equal, so the lines are not parallel.
- Perpendicular Lines: Two lines are perpendicular if the product of their slopes is [tex]\(-1\)[/tex]. Calculate the product:
[tex]\[ m_1 \times m_2 = (-0.5) \times (0.5) = -0.25 \][/tex]
Since [tex]\(-0.25\)[/tex] is not equal to [tex]\(-1\)[/tex], the lines are not perpendicular.
### Conclusion
The lines [tex]\( L_1 \)[/tex] and [tex]\( L_2 \)[/tex] are neither parallel nor perpendicular because the conditions for either relationship are not satisfied based on their slopes.
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