Let's determine the relationship between the two lines passing through the given pairs of points.
We are given the following points:
- Line [tex]\(a\)[/tex] passes through the points [tex]\((0,3)\)[/tex] and [tex]\((2,0)\)[/tex].
- Line [tex]\(b\)[/tex] passes through the points [tex]\(\left(1, -\frac{4}{3}\right)\)[/tex] and [tex]\((6, 2)\)[/tex].
To determine whether the lines are parallel, perpendicular, or neither, we first need to calculate the slopes of both lines.
### Step 1: Calculate the slope of Line [tex]\(a\)[/tex]
The formula to calculate 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:
[tex]\[ m = \frac{y_2 - y_1}{x_2 - x_1} \][/tex]
For Line [tex]\(a\)[/tex]:
- [tex]\((x_1, y_1) = (0, 3)\)[/tex]
- [tex]\((x_2, y_2) = (2, 0)\)[/tex]
Plugging these values into the slope formula:
[tex]\[ m_a = \frac{0 - 3}{2 - 0} = \frac{-3}{2} \][/tex]
[tex]\[ m_a = -1.5 \][/tex]
### Step 2: Calculate the slope of Line [tex]\(b\)[/tex]
For Line [tex]\(b\)[/tex]:
- [tex]\((x_1, y_1) = \left(1, -\frac{4}{3}\right)\)[/tex]
- [tex]\((x_2, y_2) = (6, 2)\)[/tex]
Plugging these values into the slope formula:
[tex]\[ m_b = \frac{2 - \left(-\frac{4}{3}\right)}{6 - 1} \][/tex]
First, simplify the numerator:
[tex]\[ 2 - \left(-\frac{4}{3}\right) = 2 + \frac{4}{3} = \frac{6}{3} + \frac{4}{3} = \frac{10}{3} \][/tex]
Now, calculate the slope:
[tex]\[ m_b = \frac{\frac{10}{3}}{5} = \frac{10}{3 \times 5} = \frac{10}{15} = \frac{2}{3} \][/tex]
[tex]\[ m_b = 0.6666666666666666 \][/tex]
### Step 3: Determine the relationship between the slopes
- Two lines are parallel if their slopes are equal.
- Two lines are perpendicular if the product of their slopes is [tex]\(-1\)[/tex].
#### Check if the lines are parallel:
[tex]\[ m_a = -1.5 \][/tex]
[tex]\[ m_b = 0.6666666666666666 \][/tex]
Since [tex]\( m_a \)[/tex] is not equal to [tex]\( m_b \)[/tex], the lines are not parallel.
#### Check if the lines are perpendicular:
[tex]\[ m_a \times m_b = -1.5 \times 0.6666666666666666 = -1 \][/tex]
Since the product of the slopes is [tex]\(-1\)[/tex], the lines are perpendicular.
### Conclusion
The lines passing through the points [tex]\((0,3)\)[/tex] and [tex]\((2,0)\)[/tex] for Line [tex]\(a\)[/tex], and [tex]\(\left(1, -\frac{4}{3}\right)\)[/tex] and [tex]\((6, 2)\)[/tex] for Line [tex]\(b\)[/tex], are perpendicular.
