Question

determine whether the lines ( l_1 ) and ( l_2 ) passing through the pairs of points are parallel, perpendicular, or neither. ( l_1: (1, -1), (4, 8) ) ( l_2: (1, 6), (7, 4) ) ( \bigcirc ) parallel ( \bigcirc ) perpendicular ( \bigcirc ) neither

Explanation:

Step1: Find slope of \( L_1 \)

The slope formula is \( m = \frac{y_2 - y_1}{x_2 - x_1} \). For \( L_1 \) with points \( (1, -1) \) and \( (4, 8) \), we have \( x_1 = 1, y_1 = -1, x_2 = 4, y_2 = 8 \).
\( m_1 = \frac{8 - (-1)}{4 - 1} = \frac{9}{3} = 3 \)

Step2: Find slope of \( L_2 \)

For \( L_2 \) with points \( (1, 6) \) and \( (7, 4) \), \( x_1 = 1, y_1 = 6, x_2 = 7, y_2 = 4 \).
\( m_2 = \frac{4 - 6}{7 - 1} = \frac{-2}{6} = -\frac{1}{3} \)

Step3: Check parallel or perpendicular

Parallel lines have equal slopes (\( m_1 = m_2 \)), perpendicular lines have slopes that are negative reciprocals (\( m_1 \times m_2 = -1 \)).
Check \( m_1 \times m_2 \): \( 3 \times (-\frac{1}{3}) = -1 \)

Answer:

perpendicular