Question

practice
tell whether the lines containing each pair of points are parallel.

1. (1, 3)(2, 4) and (-2, 1)(-3, 0)
2. (-4, -3)(2, 4) and (0, 2)(5, 9)
3. (3, 4)(1, 0) and (6, 8)(4, 4)
4. (5, -2)(3, 2) and (6, -1)(4, -3)

tell whether the lines containing each pair of points are perpendicular.

1. (4, 5)(1, 3) and (3, 7)(5, 4)
2. (-2, -3)(2, 5) and (-4, -4)(-6, -3)

Explanation:

Step1: Recall slope - formula

The slope formula 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 slopes for parallel - pairs

For pair 1:

For points $(1,3)$ and $(2,4)$: $m_1=\frac{4 - 3}{2 - 1}=1$.
For points $(-2,1)$ and $(-3,0)$: $m_2=\frac{0 - 1}{-3+2}=1$. Since $m_1 = m_2$, the lines are parallel.

For pair 2:

For points $(-4,-3)$ and $(2,4)$: $m_3=\frac{4 + 3}{2+4}=\frac{7}{6}$.
For points $(0,2)$ and $(5,9)$: $m_4=\frac{9 - 2}{5 - 0}=\frac{7}{5}$. Since $m_3
eq m_4$, the lines are not parallel.

For pair 3:

For points $(3,4)$ and $(1,0)$: $m_5=\frac{0 - 4}{1 - 3}=2$.
For points $(6,8)$ and $(4,4)$: $m_6=\frac{4 - 8}{4 - 6}=2$. Since $m_5 = m_6$, the lines are parallel.

For pair 4:

For points $(5,-2)$ and $(3,2)$: $m_7=\frac{2 + 2}{3 - 5}=-2$.
For points $(6,-1)$ and $(4,-3)$: $m_8=\frac{-3 + 1}{4 - 6}=1$. Since $m_7
eq m_8$, the lines are not parallel.

Step3: Calculate slopes for perpendicular - pairs

For pair 5:

For points $(4,5)$ and $(1,3)$: $m_9=\frac{3 - 5}{1 - 4}=\frac{2}{3}$.
For points $(3,7)$ and $(5,4)$: $m_{10}=\frac{4 - 7}{5 - 3}=-\frac{3}{2}$. Since $m_9\times m_{10}=- 1$, the lines are perpendicular.

For pair 6:

For points $(-2,-3)$ and $(2,5)$: $m_{11}=\frac{5 + 3}{2+2}=2$.
For points $(-4,-4)$ and $(-6,-3)$: $m_{12}=\frac{-3 + 4}{-6 + 4}=-\frac{1}{2}$. Since $m_{11}\times m_{12}=-1$, the lines are perpendicular.

Answer:

1. Parallel
2. Not parallel
3. Parallel
4. Not parallel
5. Perpendicular
6. Perpendicular