Practice with Parallel and Perpendicular Lines
buttons.Practice Problems
Answers
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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.
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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.
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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.
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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.
