Line Relations C9B687

1. **Problem Statement:** We need to determine if certain pairs of lines are perpendicular or parallel by comparing their slopes. 2. **Formulas and Rules:** - The slope $m$ of a line passing through points $(x_1, y_1)$ and $(x_2, y_2)$ is given by: $$m = \frac{y_2 - y_1}{x_2 - x_1}$$ - Two lines are **perpendicular** if the product of their slopes is $-1$: $$m_1 \times m_2 = -1$$ - Two lines are **parallel** if their slopes are equal: $$m_1 = m_2$$ --- ### Example 3: Check Perpendicularity 3. **Find slope of line $j$:** - Points: $(-1, 5)$ and $(1, 2)$ - Calculate slope: $$m_j = \frac{2 - 5}{1 - (-1)} = \frac{-3}{2} = -\frac{3}{2}$$ 4. **Find slope of line $k$:** - Points: $(-3, 2)$ and $(0, 4)$ - Calculate slope: $$m_k = \frac{4 - 2}{0 - (-3)} = \frac{2}{3}$$ 5. **Check perpendicularity:** - Multiply slopes: $$m_j \times m_k = -\frac{3}{2} \times \frac{2}{3} = -\frac{3 \times 2}{2 \times 3} = -\frac{\cancel{3} \times \cancel{2}}{\cancel{2} \times \cancel{3}} = -1$$ - Since product is $-1$, lines $j$ and $k$ are perpendicular. --- ### Example 2: Check Parallelism 6. **Find slope of line $k$:** - Points: $(-2, 2)$ and $(2, -3)$ - Calculate slope: $$m_k = \frac{-3 - 2}{2 - (-2)} = \frac{-5}{4} = -\frac{5}{4}$$ 7. **Find slope of line $n$:** - Points: $(1, 2)$ and $(4, -2)$ - Calculate slope: $$m_n = \frac{-2 - 2}{4 - 1} = \frac{-4}{3} = -\frac{4}{3}$$ 8. **Check parallelism:** - Compare slopes: $$m_k = -\frac{5}{4} \neq m_n = -\frac{4}{3}$$ - Since slopes are not equal, lines $k$ and $n$ are not parallel. --- ### Try It! 3a: Are lines $h$ and $\ell$ perpendicular? 9. **Find slope of line $h$:** - Points: $(-4, 0)$ and $(4, -5)$ - Calculate slope: $$m_h = \frac{-5 - 0}{4 - (-4)} = \frac{-5}{8} = -\frac{5}{8}$$ 10. **Find slope of line $\ell$:** - Points: $(0, -5)$ and $(4, 5)$ - Calculate slope: $$m_\ell = \frac{5 - (-5)}{4 - 0} = \frac{10}{4} = \frac{5}{2}$$ 11. **Check perpendicularity:** - Multiply slopes: $$m_h \times m_\ell = -\frac{5}{8} \times \frac{5}{2} = -\frac{25}{16} \neq -1$$ - Lines $h$ and $\ell$ are not perpendicular. --- ### Try It! 3b: Are lines $k$ and $m$ perpendicular? 12. **Find slope of line $k$:** - Points: $(-3, 2)$ and $(0, 4)$ (from earlier) - Slope: $$m_k = \frac{4 - 2}{0 - (-3)} = \frac{2}{3}$$ 13. **Find slope of line $m$:** - Points: $(0, 5)$ and $(2, -5)$ - Calculate slope: $$m_m = \frac{-5 - 5}{2 - 0} = \frac{-10}{2} = -5$$ 14. **Check perpendicularity:** - Multiply slopes: $$m_k \times m_m = \frac{2}{3} \times (-5) = -\frac{10}{3} \neq -1$$ - Lines $k$ and $m$ are not perpendicular. --- ### Try It! 2: Are lines $m$ and $q$ parallel? 15. **Find slope of line $m$:** - Points: $(0, 5)$ and $(2, -5)$ (from above) - Slope: $$m_m = -5$$ 16. **Find slope of line $q$:** - Points: (approximate from description, assume points $(0, 0)$ and $(4, 10)$ for upward slope) - Calculate slope: $$m_q = \frac{10 - 0}{4 - 0} = \frac{10}{4} = \frac{5}{2}$$ 17. **Check parallelism:** - Compare slopes: $$m_m = -5 \neq m_q = \frac{5}{2}$$ - Lines $m$ and $q$ are not parallel. --- **Final answers:** - Lines $j$ and $k$ are perpendicular. - Lines $k$ and $n$ are not parallel. - Lines $h$ and $\ell$ are not perpendicular. - Lines $k$ and $m$ are not perpendicular. - Lines $m$ and $q$ are not parallel.