Answer :
To determine the relationship between the lines passing through the given points, we need to calculate the slopes of each line and then compare them. Here's how we do it:
### Step 1: Calculate the slope of Line [tex]\( a \)[/tex]
The coordinates for Line [tex]\( a \)[/tex] are [tex]\((2, 3)\)[/tex] and [tex]\((-2, 4)\)[/tex]. The slope [tex]\( m \)[/tex] of a line passing through points [tex]\((x_1, y_1)\)[/tex] and [tex]\((x_2, y_2)\)[/tex] is given by:
[tex]\[ m = \frac{y_2 - y_1}{x_2 - x_1} \][/tex]
Plugging in the coordinates for Line [tex]\( a \)[/tex]:
[tex]\[ m_a = \frac{4 - 3}{-2 - 2} = \frac{1}{-4} = -0.25 \][/tex]
### Step 2: Calculate the slope of Line [tex]\( b \)[/tex]
The coordinates for Line [tex]\( b \)[/tex] are [tex]\((1, 5)\)[/tex] and [tex]\((7, 3)\)[/tex]. Using the slope formula:
[tex]\[ m_b = \frac{3 - 5}{7 - 1} = \frac{-2}{6} = -\frac{1}{3} \][/tex]
### Step 3: Compare the slopes
We now have the slopes of both lines:
- Slope of Line [tex]\( a \)[/tex]: [tex]\( -0.25 \)[/tex]
- Slope of Line [tex]\( b \)[/tex]: [tex]\( -\frac{1}{3} \)[/tex]
To determine if the lines are parallel, perpendicular, or neither:
- Parallel lines have equal slopes: [tex]\( m_a = m_b \)[/tex]
- Perpendicular lines have slopes that are negative reciprocals of each other: [tex]\( m_a \times m_b = -1 \)[/tex]
Let's check if the slopes are equal (for parallel lines):
[tex]\[ -0.25 \ne -\frac{1}{3} \][/tex]
The slopes are not equal, so the lines are not parallel.
Next, let's check if the slopes are negative reciprocals (for perpendicular lines):
[tex]\[ -0.25 \times -\frac{1}{3} = \frac{1}{12} \ne -1 \][/tex]
Since the product of the slopes is not [tex]\(-1\)[/tex], the lines are not perpendicular.
### Conclusion
Since the lines are neither parallel nor perpendicular, we conclude that the lines are neither.
Thus, the answer is:
[tex]\[ \text{Neither} \][/tex]
### Step 1: Calculate the slope of Line [tex]\( a \)[/tex]
The coordinates for Line [tex]\( a \)[/tex] are [tex]\((2, 3)\)[/tex] and [tex]\((-2, 4)\)[/tex]. The slope [tex]\( m \)[/tex] of a line passing through points [tex]\((x_1, y_1)\)[/tex] and [tex]\((x_2, y_2)\)[/tex] is given by:
[tex]\[ m = \frac{y_2 - y_1}{x_2 - x_1} \][/tex]
Plugging in the coordinates for Line [tex]\( a \)[/tex]:
[tex]\[ m_a = \frac{4 - 3}{-2 - 2} = \frac{1}{-4} = -0.25 \][/tex]
### Step 2: Calculate the slope of Line [tex]\( b \)[/tex]
The coordinates for Line [tex]\( b \)[/tex] are [tex]\((1, 5)\)[/tex] and [tex]\((7, 3)\)[/tex]. Using the slope formula:
[tex]\[ m_b = \frac{3 - 5}{7 - 1} = \frac{-2}{6} = -\frac{1}{3} \][/tex]
### Step 3: Compare the slopes
We now have the slopes of both lines:
- Slope of Line [tex]\( a \)[/tex]: [tex]\( -0.25 \)[/tex]
- Slope of Line [tex]\( b \)[/tex]: [tex]\( -\frac{1}{3} \)[/tex]
To determine if the lines are parallel, perpendicular, or neither:
- Parallel lines have equal slopes: [tex]\( m_a = m_b \)[/tex]
- Perpendicular lines have slopes that are negative reciprocals of each other: [tex]\( m_a \times m_b = -1 \)[/tex]
Let's check if the slopes are equal (for parallel lines):
[tex]\[ -0.25 \ne -\frac{1}{3} \][/tex]
The slopes are not equal, so the lines are not parallel.
Next, let's check if the slopes are negative reciprocals (for perpendicular lines):
[tex]\[ -0.25 \times -\frac{1}{3} = \frac{1}{12} \ne -1 \][/tex]
Since the product of the slopes is not [tex]\(-1\)[/tex], the lines are not perpendicular.
### Conclusion
Since the lines are neither parallel nor perpendicular, we conclude that the lines are neither.
Thus, the answer is:
[tex]\[ \text{Neither} \][/tex]
