Question

suppose we are given the following.
line 1 passes through (4, -1) and (8, 0).
line 2 passes through (2, -6) and (0, 2).
line 3 passes through (3, -6) and (6, -7).
(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: parallel perpendicular neither
line 1 and line 3: parallel perpendicular neither
line 2 and line 3: parallel perpendicular neither

Explanation:

Step1: Recall 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 \((4, -1)\) and \((8, 0)\).
\( m_1 = \frac{0 - (-1)}{8 - 4} = \frac{1}{4} \).

Step3: Calculate slope of Line 2

Line 2 passes through \((2, -6)\) and \((0, 2)\).
\( m_2 = \frac{2 - (-6)}{0 - 2} = \frac{8}{-2} = -4 \).

Step4: Calculate slope of Line 3

Line 3 passes through \((3, -6)\) and \((6, -7)\).
\( m_3 = \frac{-7 - (-6)}{6 - 3} = \frac{-1}{3} = -\frac{1}{3} \).

Step5: Analyze Line 1 and Line 2

Check if \( m_1 \times m_2 = -1 \) (perpendicular) or \( m_1 = m_2 \) (parallel).
\( \frac{1}{4} \times (-4) = -1 \), so they are perpendicular.

Step6: Analyze Line 1 and Line 3

Check slopes: \( \frac{1}{4}
eq -\frac{1}{3} \), and \( \frac{1}{4} \times (-\frac{1}{3}) = -\frac{1}{12}
eq -1 \). So neither.

Step7: Analyze Line 2 and Line 3

Check slopes: \( -4
eq -\frac{1}{3} \), and \( -4 \times (-\frac{1}{3}) = \frac{4}{3}
eq -1 \). So neither.

Answer:

(a)
Slope of Line 1: \(\frac{1}{4}\)
Slope of Line 2: \(-4\)
Slope of Line 3: \(-\frac{1}{3}\)

(b)
Line 1 and Line 2: Perpendicular
Line 1 and Line 3: Neither
Line 2 and Line 3: Neither