To determine whether the lines are parallel, perpendicular, or neither, we need to calculate the slopes of both lines and compare them.
### Step-by-Step Solution:
#### Step 1: Calculate the slope for Line [tex]\( a \)[/tex]:
Given the points [tex]\((6,2)\)[/tex] and [tex]\((9,3)\)[/tex] for Line [tex]\( a \)[/tex]:
The slope [tex]\( m_a \)[/tex] is calculated using the slope formula:
[tex]\[ m_a = \frac{y_2 - y_1}{x_2 - x_1} \][/tex]
Substitute the coordinates:
[tex]\[ m_a = \frac{3 - 2}{9 - 6} \][/tex]
[tex]\[ m_a = \frac{1}{3} \][/tex]
[tex]\[ m_a = 0.3333333333333333 \][/tex]
#### Step 2: Calculate the slope for Line [tex]\( b \)[/tex]:
Given the points [tex]\((1,11)\)[/tex] and [tex]\((3,5)\)[/tex] for Line [tex]\( b \)[/tex]:
The slope [tex]\( m_b \)[/tex] is calculated using the same slope formula:
[tex]\[ m_b = \frac{y_2 - y_1}{x_2 - x_1} \][/tex]
Substitute the coordinates:
[tex]\[ m_b = \frac{5 - 11}{3 - 1} \][/tex]
[tex]\[ m_b = \frac{-6}{2} \][/tex]
[tex]\[ m_b = -3 \][/tex]
#### Step 3: Compare the slopes:
- Parallel Lines: Lines are parallel if their slopes are equal, i.e., [tex]\( m_a = m_b \)[/tex].
- Perpendicular Lines: Lines are perpendicular if the product of their slopes is [tex]\(-1\)[/tex], i.e., [tex]\( m_a \times m_b = -1 \)[/tex].
- Neither: If neither condition is satisfied, the lines are neither parallel nor perpendicular.
Let's check:
Slopes:
[tex]\[ m_a = 0.3333333333333333 \][/tex]
[tex]\[ m_b = -3 \][/tex]
Calculate the product of the slopes:
[tex]\[ m_a \times m_b = 0.3333333333333333 \times (-3) = -1 \][/tex]
Since the product of the slopes [tex]\( m_a \)[/tex] and [tex]\( m_b \)[/tex] is [tex]\(-1\)[/tex], the lines are perpendicular.
### Conclusion:
The lines are Perpendicular.
Thus, the correct selection is:
Perpendicular
