Question

Help im not that smart

Answer 1

keeping in mind that perpendicular lines have negative reciprocal slopes, and that parallel lines have exactly the same slope, let's check for the slope of each of those lines.

`(\stackrel{x_1}{-2}~,~\stackrel{y_1}{0})\qquad (\stackrel{x_2}{1}~,~\stackrel{y_2}{6}) \\\\\\ \stackrel{slope}{m}\implies \cfrac{\stackrel{\textit{\large rise}} {\stackrel{y_2}{6}-\stackrel{y1}{0}}}{\underset{\textit{\large run}} {\underset{x_2}{1}-\underset{x_1}{(-2)}}} \implies \cfrac{ 6 }{1 +2} \implies \cfrac{ 6 }{ 3 } \implies 2\qquad \textit{\Huge a} \\\\[-0.35em] ~\dotfill`

`(\stackrel{x_1}{-3}~,~\stackrel{y_1}{5})\qquad (\stackrel{x_2}{3}~,~\stackrel{y_2}{1}) \\\\\\ \stackrel{slope}{m}\implies \cfrac{\stackrel{\textit{\large rise}} {\stackrel{y_2}{1}-\stackrel{y1}{5}}}{\underset{\textit{\large run}} {\underset{x_2}{3}-\underset{x_1}{(-3)}}} \implies \cfrac{ -4 }{3 +3} \implies \cfrac{ -4 }{ 6 } \implies -\cfrac{2}{3}\qquad \textit{\Huge b} \\\\[-0.35em] ~\dotfill`

`(\stackrel{x_1}{1}~,~\stackrel{y_1}{1})\qquad (\stackrel{x_2}{4}~,~\stackrel{y_2}{7}) \\\\\\ \stackrel{slope}{m}\implies \cfrac{\stackrel{\textit{\large rise}} {\stackrel{y_2}{7}-\stackrel{y1}{1}}}{\underset{\textit{\large run}} {\underset{x_2}{4}-\underset{x_1}{1}}} \implies \cfrac{ 6 }{ 3 } \implies 2\qquad \textit{\Huge c} \\\\[-0.35em] ~\dotfill\\\\ \stackrel{slope}{\textit{\Huge a}}~~ = ~~\stackrel{slope}{\textit{\Huge c}}\qquad \impliedby parallel`

Answer 2

Lines a and c are parallel lines since they have the same slope.

None of the lines are perpendicular because none of their slopes are negative reciprocals of each other, which is a condition for lines to be perpendicular.

Explanation:

To determine whether the given lines are parallel or perpendicular, we can examine their slopes.

The slope of a line passing through two points (x₁, y₁) and (x₂, y₂) is given by the slope formula:

`m = (y_2 - y_1)/(x_2 - x_1)`

Let's calculate the slopes for the three lines by substituting the given points into the slope formula.

Slope of line a

Line a passes through points (-2, 0) and (1, 6):

`m_a = (6 - 0)/(1 - (-2)) = (6)/(3) = 2`

Slope of line b

Line b passes through points (-3, 5) and (3, 1):

`m_b = (1 - 5)/(3 - (-3)) = (-4)/(6) = -(2)/(3)`

Slope of line c

Line c passes through points (1, 1) and (4, 7):

`m_c = (7 - 1)/(4 - 1) = (6)/(3) = 2`

Therefore:

• Line a has a slope of 2.
• Line b has a slope of -2/3.
• Line c has a slope of 2.

Parallel lines have the same slope. Therefore, lines a and c are parallel lines since they both have a slope of 2.

Perpendicular lines have slopes that are negative reciprocals of each other. Since none of the lines have slopes that satisfy this condition, none of the lines are perpendicular.