Expand menu menu_open Minimize Start chapters Home History history History expand_more
{{ item.displayTitle }}
navigate_next
No history yet!
Progress & Statistics equalizer Progress expand_more
Student
navigate_next
Teacher
navigate_next
{{ filterOption.label }}
{{ item.displayTitle }}
{{ item.subject.displayTitle }}
arrow_forward
No results
{{ searchError }}
search
menu_open
{{ courseTrack.displayTitle }}
{{ statistics.percent }}% Sign in to view progress
{{ printedBook.courseTrack.name }} {{ printedBook.name }}
search Use offline Tools apps
Login account_circle menu_open

Proving Relationships of Parallel and Perpendicular Lines

Proving Relationships of Parallel and Perpendicular Lines 1.2 - Solution

arrow_back Return to Proving Relationships of Parallel and Perpendicular Lines

Parallel lines have the same slope but different y-y\text{-}intercepts. Perpendicular lines have slopes that are opposite reciprocals of one another. Therefore, to determine if the lines are parallel, perpendicular, or neither, we first need to find their individual slopes using the Slope Formula.

Line Points y2y1x2x1\dfrac{y_2-y_1}{x_2-x_1} Slope
pp (-2,4)&(2,3)({\color{#0000FF}{\text{-} 2,4}}) \, \& \, ({\color{#009600}{2,3}}) 342(-2)\dfrac{{\color{#009600}{3}}-{\color{#0000FF}{4}}}{{\color{#009600}{2}}-({\color{#0000FF}{\text{-} 2}})} -14\text{-}\dfrac{1}{4}
qq (-3,1)&(3,-1)({\color{#0000FF}{\text{-} 3,1}}) \, \& \, ({\color{#009600}{3,\text{-} 1}}) -113(-3)\dfrac{{\color{#009600}{\text{-} 1}}-{\color{#0000FF}{1}}}{{\color{#009600}{3}}-({\color{#0000FF}{\text{-} 3}})} -13\text{-}\dfrac{1}{3}
\ell (0,-2)&(2,4)({\color{#0000FF}{0,\text{-} 2}}) \, \& \, ({\color{#009600}{2,4}}) 4(-2)20\dfrac{{\color{#009600}{4}}-({\color{#0000FF}{\text{-} 2}})}{{\color{#009600}{2}}-{\color{#0000FF}{0}}} 33
mm (-2,0)&(0,6)({\color{#0000FF}{\text{-} 2,0}}) \, \& \, ({\color{#009600}{0,6}}) 600(-2)\dfrac{{\color{#009600}{6}}-{\color{#0000FF}{0}}}{{\color{#009600}{0}}-({\color{#0000FF}{\text{-} 2}})} 33

Lines \ell and mm have the same slope, but different y-y\text{-}intercepts. Therefore, they are parallel. Meanwhile, the slopes of lines q&q\,\&\, \ell and q&mq\,\&\,m are opposite reciprocals, so they are perpendicular. The line pp is neither parallel nor perpendicular to lines q,q, ,\ell, and m.m.