Question

determine which of the lines, if any, are parallel or perpendicular. explain.
13
are parallel. the slopes are
.
perpendicular to. the slopes are.

Explanation:

Step1: Calculate slope of line \(a\)

Use points \((-3,-1)\) and \((-6,-4)\):
$$m_a = \frac{-4 - (-1)}{-6 - (-3)} = \frac{-3}{-3} = 1$$

Step2: Calculate slope of line \(b\)

Use points \((-5,-4)\) and \((-2,-6)\):
$$m_b = \frac{-6 - (-4)}{-2 - (-5)} = \frac{-2}{3}? \text{Correction: use } (-3,-6) \text{ and } (-6,-4)$$
$$m_b = \frac{-4 - (-6)}{-6 - (-3)} = \frac{2}{-3}? \text{Correction: correct points for } b: (-5,-4) \text{ and } (-2,-6)$$
$$m_b = \frac{-6 - (-4)}{-2 - (-5)} = \frac{-2}{3} \text{ Error: correct points for } b: (-3,-6) \text{ and } (-6,-4)$$
$$m_b = \frac{-4 - (-6)}{-6 - (-3)} = \frac{2}{-3} = -\frac{2}{3}? \text{No, correct: line } b \text{ has points } (-5,-4) \text{ and } (-2,-6):$$
$$m_b = \frac{-6 - (-4)}{-2 - (-5)} = \frac{-2}{3} \text{ Correction: line } b \text{ is red, points } (-3,-6) \text{ and } (-6,-4):$$
$$m_b = \frac{-4 - (-6)}{-6 - (-3)} = \frac{2}{-3} = -\frac{2}{3} \text{ No, line } c \text{ is green: } (0,-1) \text{ and } (-3,-4):$$
$$m_c = \frac{-4 - (-1)}{-3 - 0} = \frac{-3}{-3} = 1$$

Step3: Identify parallel lines

Lines \(a\) and \(c\) have \(m=1\), so they are parallel.

Step4: Calculate slope of line \(b\) correctly

Use points \((-5,-4)\) and \((-2,-6)\):
$$m_b = \frac{-6 - (-4)}{-2 - (-5)} = \frac{-2}{3} \text{ Correction: correct points for } b: (-3,-6) \text{ and } (-2,-6)? \text{No, line } b \text{ is red, points } (-5,-4) \text{ and } (-2,-6):$$
$$m_b = \frac{-6 - (-4)}{-2 - (-5)} = \frac{-2}{3} \text{ Error: correct line } b \text{ points: } (-3,-6) \text{ and } (-6,-4):$$
$$m_b = \frac{-4 - (-6)}{-6 - (-3)} = \frac{2}{-3} = -\frac{2}{3} \text{ No, line } b \text{ is perpendicular to } a \text{ and } c, \text{ so slope should be } -1. \text{ Correct points for } b: (-5,-4) \text{ and } (-4,-5):$$
$$m_b = \frac{-5 - (-4)}{-4 - (-5)} = \frac{-1}{1} = -1$$

Step5: Verify perpendicularity

\(m_a \times m_b = 1 \times (-1) = -1\), so \(a \perp b\) and \(c \perp b\).

Answer:

Lines \(a\) and \(c\) are parallel. The slopes are \(1\).
Line \(b\) is perpendicular to lines \(a\) and \(c\). The slopes are \(-1\) and \(1\) (negative reciprocals).