Question

How to know if points are parallel, perpendicular or neither

line 1 (7,3) and (8,1)
line 2 (-1,-2) and (-3, -1)

Answer 1

well, simply let's take a peek of their slopes, keeping in mind that perpendicular lines have negative reciprocal slopes, and that parallel lines have exactly the same slope

`\stackrel{ \textit{\LARGE line 1} }{(\stackrel{x_1}{7}~,~\stackrel{y_1}{3})\qquad (\stackrel{x_2}{8}~,~\stackrel{y_2}{1})} ~\hfill~ \stackrel{slope}{m}\implies \cfrac{\stackrel{\textit{\large rise}} {\stackrel{y_2}{1}-\stackrel{y1}{3}}}{\underset{\textit{\large run}} {\underset{x_2}{8}-\underset{x_1}{7}}} \implies \cfrac{ -2 }{ 1 } \implies -2 \\\\[-0.35em] ~\dotfill`

`\stackrel{ \textit{\LARGE line 2} }{(\stackrel{x_1}{-1}~,~\stackrel{y_1}{-2})\qquad (\stackrel{x_2}{-3}~,~\stackrel{y_2}{-1})} \\\\\\ \stackrel{slope}{m}\implies \cfrac{\stackrel{\textit{\large rise}} {\stackrel{y_2}{-1}-\stackrel{y1}{(-2)}}}{\underset{\textit{\large run}} {\underset{x_2}{-3}-\underset{x_1}{(-1)}}} \implies \cfrac{-1 +2}{-3 +1} \implies \cfrac{ 1 }{ -2 } \implies -\cfrac{1}{2} \\\\[-0.35em] ~\dotfill\\\\ \textit{\LARGE lines 1 {\small and} 2}\textit{ are neither perpendicular or parallel}`