Practice with Parallel and Perpendicular Lines

Practice Problems

1. Are the lines in the figure parallel, perpendicular, or neither?

   
   

   Choose 1 answer:
   

   1. Parallel
   2. Perpendicular

   3. Neither

2. One line passes through the points ‍(-2,1) and ‍(4, 9). Another line passes through points ‍(-3, 8) and (5, 2).

   Are the lines parallel, perpendicular, or neither?
   

   Choose 1 answer:
   

   1. Parallel
   2. Perpendicular
   3. Neither

3. Are the lines in the figure parallel, perpendicular, or neither?

   
   

   Choose 1 answer:
   

   1. Parallel
   2. Perpendicular

   3. Neither

4. One line passes through the points ‍(-7, -4) and ‍(5, 4). Another line passes through points ‍(-4, 6) and (6, -9).

   Are the lines parallel, perpendicular, or neither?
   

   Choose 1 answer:
   

   1. Parallel
   2. Perpendicular
   3. Neither

Source: Khan Academy, https://www.khanacademy.org/math/geometry/hs-geo-analytic-geometry/hs-geo-parallel-perpendicular-lines/e/classifying-lines-as-parallel--perpendicular--or-neither
This work is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 3.0 License.

Answers

1. The lines clearly aren't parallel, but are they perpendicular?

   Perpendicular lines have slopes that are negative reciprocals of each other.

   Step 1: Finding the slope of each line

   Slope is the ratio of the vertical and horizontal changes between any two points on a line.

   The slope of line a:
   

   \(\dfrac{\Delta y}{\Delta x} = {\dfrac{7}{10}}\)

   The slope of line b:
   

   \(\dfrac{\Delta y}{\Delta x} = {\dfrac{-12}{9}} = -\dfrac{4}{3}\)

   
   

   Step 2: Comparing the slopes
   

   The negative reciprocal of \(\frac{7}{10}\) is \(-\frac{10}{7}\), not \(-\frac{4}{3}\) , so the lines are not perpendicular.

   Answer

   The lines are neither parallel nor perpendicular.

2. Slopes of parallel and perpendicular lines

   Parallel lines have the same slope.
   

   Perpendicular lines have slopes that are negative reciprocals of each other.

   Step 1: Finding the slope of the first line

   \(\begin{aligned}
   \dfrac{\Delta y}{\Delta x} &= \dfrac{y_2 - y_1}{x_2 - x_1} \\\\\\
   &= \dfrac{9-1}{4-(-2)} \\\\\\
   &= \dfrac{8}{6} \\\\\\
   &= {\dfrac{4}{3}}
   \end{aligned}\)

   The ‍y-value increases as the x-value increases, so it makes sense that this line has a positive slope.

   Step 2: Finding the slope of the second line
   
   \(\begin{aligned}
   \dfrac{\Delta y}{\Delta x} &= \dfrac{y_2 - y_1}{x_2 - x_1} \\\\\\
   &= \dfrac{2-8}{5-(-3)} \\\\\\
   &= \dfrac{-6}{8} \\\\\\
   &= {-\dfrac{3}{4}}
   \end{aligned}\)

   The y-value decreases as the x-value increases, so it makes sense that this line has a negative slope.

   Step 3: Comparing the slopes
   

   The negative reciprocal of \(\frac{4}{3}\) is \(-\frac{3}{4}\), so the lines are perpendicular.

   Answer

   The lines are perpendicular.

   
   

3. The lines clearly aren't parallel, but are they perpendicular?

   Perpendicular lines have slopes that are negative reciprocals of each other.

   Step 1: Finding the slope of each line

   Slope is the ratio of the vertical and horizontal changes between any two points on a line.

   The slope of line a:
   

   \(\dfrac{\Delta y}{\Delta x} = {\dfrac{7}{10}}\)

   The slope of line b:
   

   \(\dfrac{\Delta y}{\Delta x} = {\dfrac{14}{12}} = \dfrac{7}{6}\)

   
   

   Step 2: Comparing the slopes
   

   The negative reciprocal of \(\frac{7}{6}\) is \(-\frac{6}{7}\), not \(-\frac{7}{6}\) , so the lines are not perpendicular.

   Answer

   The lines are neither parallel nor perpendicular.

4. Slopes of parallel and perpendicular lines

   Parallel lines have the same slope.
   

   Perpendicular lines have slopes that are negative reciprocals of each other.

   Step 1: Finding the slope of the first line

   \(\begin{aligned}
   \dfrac{\Delta y}{\Delta x} &= \dfrac{y_2 - y_1}{x_2 - x_1} \\\\\\
   &= \dfrac{4-(-4)}{5-(-7)} \\\\\\
   &= \dfrac{8}{12} \\\\\\
   &= {\dfrac{2}{3}}
   \end{aligned}\)

   The ‍y-value increases as the x-value increases, so it makes sense that this line has a positive slope.

   Step 2: Finding the slope of the second line
   
   \(\begin{aligned}
   \dfrac{\Delta y}{\Delta x} &= \dfrac{y_2 - y_1}{x_2 - x_1} \\\\\\
   &= \dfrac{-9-6}{6-(-4)} \\\\\\
   &= \dfrac{-15}{10} \\\\\
   &= {-\dfrac{3}{2}}
   \end{aligned}\)

   The y-value decreases as the x-value increases, so it makes sense that this line has a negative slope.

   Step 3: Comparing the slopes
   

   The negative reciprocal of \(\frac{2}{3}\) is \(-\frac{3}{2}\), so the lines are perpendicular.

   Answer

   The lines are perpendicular.