Question

suppose we are given the following.
line 1 passes through $(-3, -4)$ and $(3, 0)$.
line 2 passes through $(-2, -8)$ and $(2, -2)$.
line 3 passes through $(4, 8)$ and $(0, 2)$.
(a) find the slope of each line.
slope of line 1:
slope of line 2:
slope of line 3:
(b) for each pair of lines, determine whether they are parallel, perpendicular, or neither.
line 1 and line 2: $circ$ parallel $circ$ perpendicular $circ$ neither
line 1 and line 3: $circ$ parallel $circ$ perpendicular $circ$ neither
line 2 and line 3: $circ$ parallel $circ$ perpendicular $circ$ neither

Explanation:

Part (a)

Step1: Recall the slope formula

The slope \( m \) between two points \( (x_1, y_1) \) and \( (x_2, y_2) \) is \( m=\frac{y_2 - y_1}{x_2 - x_1} \).

Step2: Calculate slope of Line 1

Line 1 passes through \( (-3, -4) \) and \( (3, 0) \). Let \( (x_1, y_1)=(-3, -4) \) and \( (x_2, y_2)=(3, 0) \).
\( m_1=\frac{0 - (-4)}{3 - (-3)}=\frac{0 + 4}{3 + 3}=\frac{4}{6}=\frac{2}{3} \)

Step3: Calculate slope of Line 2

Line 2 passes through \( (-2, -8) \) and \( (2, -2) \). Let \( (x_1, y_1)=(-2, -8) \) and \( (x_2, y_2)=(2, -2) \).
\( m_2=\frac{-2 - (-8)}{2 - (-2)}=\frac{-2 + 8}{2 + 2}=\frac{6}{4}=\frac{3}{2} \)

Step4: Calculate slope of Line 3

Line 3 passes through \( (4, 8) \) and \( (0, 2) \). Let \( (x_1, y_1)=(4, 8) \) and \( (x_2, y_2)=(0, 2) \).
\( m_3=\frac{2 - 8}{0 - 4}=\frac{-6}{-4}=\frac{3}{2} \)

• Two lines are parallel if their slopes are equal (\( m_1 = m_2 \)).
• Two lines are perpendicular if the product of their slopes is \(- 1\) (\( m_1\times m_2=-1 \)).

Line 1 and Line 2:

\( m_1=\frac{2}{3} \), \( m_2=\frac{3}{2} \).
Product: \( \frac{2}{3}\times\frac{3}{2}=1
eq - 1 \), and \( \frac{2}{3}
eq\frac{3}{2} \). So, neither.

Line 1 and Line 3:

\( m_1=\frac{2}{3} \), \( m_3=\frac{3}{2} \).
Product: \( \frac{2}{3}\times\frac{3}{2}=1
eq - 1 \), and \( \frac{2}{3}
eq\frac{3}{2} \). So, neither.

Line 2 and Line 3:

\( m_2=\frac{3}{2} \), \( m_3=\frac{3}{2} \).
Since \( m_2 = m_3 \), they are parallel.

Answer:

Slope of Line 1: \(\frac{2}{3}\)
Slope of Line 2: \(\frac{3}{2}\)
Slope of Line 3: \(\frac{3}{2}\)

Part (b)