Question

determine which of the lines are parallel and which of the lines are perpendicular. select all of the statements that are true.
line a passes through (2, -4) and (14, -7)
line b passes through (-1, 10) and (1, 2)
line c passes through (-6, -5) and (-5, -9)
line d passes through (2, 0) and (-10, -3)
☐ a∥b

Explanation:

Step1: Calculate slope of line a

Slope formula: $m = \frac{y_2 - y_1}{x_2 - x_1}$
$m_a = \frac{-7 - (-4)}{14 - 2} = \frac{-3}{12} = -\frac{1}{4}$

Step2: Calculate slope of line b

$m_b = \frac{2 - 10}{1 - (-1)} = \frac{-8}{2} = -4$

Step3: Calculate slope of line c

$m_c = \frac{-9 - (-5)}{-5 - (-6)} = \frac{-4}{1} = -4$

Step4: Calculate slope of line d

$m_d = \frac{-3 - 0}{-10 - 2} = \frac{-3}{-12} = \frac{1}{4}$

Step5: Check parallelism (equal slopes)

$m_b = m_c = -4$ → b∥c; $m_a = -\frac{1}{4}$, $m_d = \frac{1}{4}$ (not equal)

Step6: Check perpendicularity (product=-1)

$m_a \times m_b = -\frac{1}{4} \times (-4) = 1 ≠ -1$; $m_a \times m_c = -\frac{1}{4} \times (-4) =1 ≠-1$; $m_a \times m_d = -\frac{1}{4} \times \frac{1}{4}=-1/16≠-1$; $m_b \times m_d=-4×1/4=-1$ → b⊥d; $m_c×m_d=-4×1/4=-1$→c⊥d

Answer:

b∥c, b⊥d, c⊥d (assuming these are the true statements; since the full options aren’t visible, the valid pairs are b&c parallel, b&d perpendicular, c&d perpendicular)