Question

suppose we are given the following.
line 1 passes through (2, 2) and (4, 7).
line 2 passes through (2, 8) and (-2, -2).
line 3 passes through (-2, 2) and (0, 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 \( (2, 2) \) and \( (4, 7) \). So, \( x_1 = 2, y_1 = 2, x_2 = 4, y_2 = 7 \).
\( m_1 = \frac{7 - 2}{4 - 2} = \frac{5}{2} \).

Step3: Calculate slope of Line 2

Line 2 passes through \( (2, 8) \) and \( (-2, -2) \). So, \( x_1 = 2, y_1 = 8, x_2 = -2, y_2 = -2 \).
\( m_2 = \frac{-2 - 8}{-2 - 2} = \frac{-10}{-4} = \frac{5}{2} \).

Step4: Calculate slope of Line 3

Line 3 passes through \( (-2, 2) \) and \( (0, 7) \). So, \( x_1 = -2, y_1 = 2, x_2 = 0, y_2 = 7 \).
\( m_3 = \frac{7 - 2}{0 - (-2)} = \frac{5}{2} \).

Step5: Analyze Line 1 and Line 2

Since \( m_1 = m_2 = \frac{5}{2} \), they are parallel.

Step6: Analyze Line 1 and Line 3

Since \( m_1 = m_3 = \frac{5}{2} \), they are parallel.

Step7: Analyze Line 2 and Line 3

Since \( m_2 = m_3 = \frac{5}{2} \), they are parallel.

Answer:

(a)
Slope of Line 1: \(\frac{5}{2}\)
Slope of Line 2: \(\frac{5}{2}\)
Slope of Line 3: \(\frac{5}{2}\)

(b)
Line 1 and Line 2: Parallel
Line 1 and Line 3: Parallel
Line 2 and Line 3: Parallel